# Quadratic Equations and Complex Numbers

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1 Quadratic Equations and Comple Numbers.1 Solving Quadratic Equations. Comple Numbers.3 Completing the Square. Using the Quadratic Formula.5 Solving Nonlinear Sstems. Quadratic Inequalities Robot-Building Competition (p. 197) Broadcast Tower (p. 189) Gannet (p. 181) SEE the Big Idea Baseball (p. 17) Electrical l Circuits (p. 158) Mathematical Thinking: Mathematicall proficient students can appl the mathematics the know to solve problems arising in everda life, societ, and the workplace.

2 Maintaining Mathematical Proficienc Simplifing Square Roots (A.11.A) Eample 1 Simplif 8. 8 = 1 >. So, subtract from 1. Factor using the greatest perfect square factor. = Product Use the Propert sign of of 1. Square Roots = Simplif. ab = a b, where a, b 0 7 Eample Simplif = 7 3 = 7 Quotient Propert of Square Roots Simplif. a a b =, where a 0 and b > 0 b Simplif the epression The integers have the same sign Factoring Special Products (A.10.E, A.10.F) Eample 3 Factor (a) and (b) The product is positive. a. = Write as a b. The integers = have ( + different )( signs. ) Difference of Two Squares Pattern So, = ( + )( ). b = ()(7) + 7 Write as a ab + b. The quotient is negative. = ( 7) Perfect Square Trinomial Pattern Factor the polnomial. So, = ( 7) ABSTRACT REASONING Determine the possible integer values of a and c for which the trinomial a c is factorable using the Perfect Square Trinomial Pattern. Eplain our reasoning. 13

3 Mathematical Thinking Mathematicall profi cient students select tools, including real objects, manipulatives, paper and pencil, and technolog as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. (A.1.C) Recognizing the Limitations of Technolog Core Concept Graphing Calculator Limitations Graphing calculators have a limited number of piels to displa the graph of a function. The result ma be an inaccurate or misleading graph. To correct this issue, use a viewing window setting based on the dimensions of the screen (in piels). Recognizing an Incorrect Graph Use a graphing calculator to draw the circle given b the equation + =.5. Begin b solving the equation for. 5 =.5 =.5 Equation of upper semicircle Equation of lower semicircle 5 5 The graphs of these two equations are shown in the first viewing window. Notice that there are two issues. First, the graph resembles an oval rather than a circle. Second, the two parts of the graph appear to have gaps between them. 5 You can correct the first issue b using a square viewing window, as shown in the second viewing window. To correct the second issue, ou need to know the dimensions of the graphing calculator screen in terms of the number of piels. For instance, for a screen that is 3 piels high and 95 piels wide, use a viewing window setting as shown at the right Monitoring Progress 1. Eplain wh the second viewing window in Eample 1 shows gaps between the upper and lower semicircles, but the third viewing window does not show gaps. Use a graphing calculator to draw an accurate graph of the equation. Eplain our choice of viewing window.. = =.5. + = = = = Chapter Quadratic Equations and Comple Numbers

6 Solving Quadratic Equations Algebraicall When solving quadratic equations using square roots, ou can use properties of square roots to write our solutions in different forms. When a radicand in the denominator of a fraction is not a perfect square, ou can multipl the fraction b an appropriate form of 1 to eliminate the radical from the denominator. This process is called rationalizing the denominator. Solving Quadratic Equations Using Square Roots ANALYZING MATHEMATICAL RELATIONSHIPS Notice that ( + 3) = 5 is of the form u = d, where u = + 3. STUDY TIP Because = 1, the value of 5 does not change when ou multipl b. Solve each equation using square roots. a. 31 = 9 b = 0 c. 5 ( + 3) = 5 a. 31 = 9 Write the equation. = 80 Add 31 to each side. = 0 Divide each side b. = ± 0 = ± 5 = ± 5 Take square root of each side. Product Propert of Square Roots Simplif. The solutions are = 5 and = 5. b = 0 Write the equation. 3 = 9 Subtract 9 from each side. = 3 Divide each side b 3. The square of a real number cannot be negative. So, the equation has no real solution. c. 5 ( + 3) = 5 Write the equation. ( + 3) = = ± 5 = 3 ± 5 = 3 ± 5 = 3 ± 5 = 3 ± 5 Multipl each side b 5. Take square root of each side. Subtract 3 from each side. Quotient Propert of Square Roots Multipl b. Simplif. The solutions are = and = 3 5. Monitoring Progress Solve the equation using square roots. Help in English and Spanish at BigIdeasMath.com = =. ( ) = 5 Section.1 Solving Quadratic Equations 17

7 When the left side of a + b + c = 0 is factorable, ou can solve the equation using the Zero-Product Propert. Core Concept Zero-Product Propert Words If the product of two epressions is zero, then one or both of the epressions equal zero. Algebra If A and B are epressions and AB = 0, then A = 0 or B = 0. Solve = 5 b factoring. Solving a Quadratic Equation b Factoring UNDERSTANDING MATHEMATICAL TERMS If a real number k is a zero of the function f() = a + b + c, then k is an -intercept of the graph of the function, and k is also a root of the equation a + b + c = 0. = 5 5 = 0 Write the equation. Write in standard form. ( 9)( + 5) = 0 Factor the polnomial. 9 = 0 or + 5 = 0 Zero-Product Propert = 9 or = 5 Solve for. The solutions are = 5 and = 9. You know the -intercepts of the graph of f () = a( p)( q) are p and q. Because the value of the function is zero when = p and when = q, the numbers p and q are also called zeros of the function. A zero of a function f is an -value for which f () = 0. Check Zero X=1.5 Y=0 8 Find the zeros of f () = Finding the Zeros of a Quadratic Function To find the zeros of the function, find the -values for which f () = = 0 Set f () equal to 0. ( 3)( ) = 0 Factor the polnomial. 3 = 0 or = 0 Zero-Product Propert = 1.5 or = Solve for. The zeros of the function are = 1.5 and =. You can check this b graphing the function. The -intercepts are 1.5 and. Monitoring Progress Solve the equation b factoring. Help in English and Spanish at BigIdeasMath.com = = Find the zero(s) of the function. 9. f () = f () = Chapter Quadratic Equations and Comple Numbers

8 Solving Real-Life Problems To find the maimum value or minimum value of a quadratic function, ou can first use factoring to write the function in intercept form f () = a( p)( q). Because the verte of the function lies on the ais of smmetr, = p + q, the maimum value or minimum value occurs at the average of the zeros p and q. Solving a Multi-Step Problem A monthl teen magazine has 8,000 subscribers when it charges \$0 per annual subscription. For each \$1 increase in price, the magazine loses about 000 subscribers. How much should the magazine charge to maimize annual revenue? What is the maimum annual revenue? Step 1 Define the variables. Let represent the price increase and R() represent the annual revenue. Step Write a verbal model. Then write and simplif a quadratic function. Annual revenue (dollars) = Number of subscribers (people) Subscription price (dollars/person) R() = (8, ) (0 + ) R() = ( ,000)( + 0) R() = 000( )( + 0) Step 3 Identif the zeros and find their average. Then find how much each subscription should cost to maimize annual revenue. The zeros of the revenue function are and 0. The average of the zeros + ( 0) is =. To maimize revenue, each subscription should cost \$0 + \$ = \$. Step Find the maimum annual revenue. R() = 000( )( + 0) = \$98,000 So, the magazine should charge \$ per subscription to maimize annual revenue. The maimum annual revenue is \$98,000. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 11. WHAT IF? The magazine initiall charges \$1 per annual subscription. How much should the magazine charge to maimize annual revenue? What is the maimum annual revenue? Section.1 Solving Quadratic Equations 19

9 Height (feet) Height of Dropped Object h 1 3 t Time (seconds) 0 sec, 00 ft 1 sec, 18 ft sec, 13 ft 3 sec, 5 ft 3.5 sec, 0 ft ANALYZING MATHEMATICAL RELATIONSHIPS In the model for the height of a dropped object, the term 1t indicates that an object has fallen 1t feet after t seconds. When an object is dropped, its height h (in feet) above the ground after t seconds can be modeled b the function h = 1t + h 0, where h 0 is the initial height (in feet) of the object. The graph of h = 1t + 00, representing the height of an object dropped from an initial height of 00 feet, is shown at the left. The model h = 1t + h 0 assumes that the force of air resistance on the object is negligible. Also, this model applies onl to objects dropped on Earth. For planets with stronger or weaker gravitational forces, different models are used. Modeling a Dropped Object For a science competition, students must design a container that prevents an egg from breaking when dropped from a height of 50 feet. a. Write a function that gives the height h (in feet) of the container after t seconds. How long does the container take to hit the ground? b. Find and interpret h(1) h(1.5). a. The initial height is 50, so the model is h = 1t Find the zeros of the function. h = 1t + 50 Write the function. 0 = 1t + 50 Substitute 0 for h. 50 = 1t Subtract 50 from each side = t Divide each side b ± = t Take square root of each side. 1 ±1.8 t Use a calculator. Reject the negative solution, 1.8, because time must be positive. The container will fall for about 1.8 seconds before it hits the ground. b. Find h(1) and h(1.5). These represent the heights after 1 and 1.5 seconds. h(1) = 1(1) + 50 = = 3 h(1.5) = 1(1.5) + 50 = 1(.5) + 50 = = 1 h(1) h(1.5) = 3 1 = 0 So, the container fell 0 feet between 1 and 1.5 seconds. You can check this b graphing the function. The points appear to be about 0 feet apart. So, the answer is reasonable. Check h (1, h(1)) (1.5, h(1.5)) 0 ft t Monitoring Progress 150 Chapter Quadratic Equations and Comple Numbers Help in English and Spanish at BigIdeasMath.com 1. WHAT IF? The egg container is dropped from a height of 80 feet. How does this change our answers in parts (a) and (b)?

10 .1 Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check 1. WRITING Eplain how to use graphing to find the roots of the equation a + b + c = 0.. DIFFERENT WORDS, SAME QUESTION Which is different? Find both answers. What are the zeros of f () = ? What are the solutions of = 0? What are the roots of 10 = 3? What is the -intercept of the graph of = ( + 5)( )? Monitoring Progress and Modeling with Mathematics In Eercises 3 1, solve the equation b graphing. (See Eample 1.) = = = 9. 8 = 7. 8 = 8. 3 = = 10. = = 1. 3 = In Eercises 13 0, solve the equation using square roots. (See Eample.) 13. s = 1 1. a = (z ) = 5 1. (p ) = ( 1) + = ( + ) 5 = r 10 = 3 r = ANALYZING RELATIONSHIPS Which equations have roots that are equivalent to the -intercepts of the graph shown? A 8 = 0 B 0 = ( + )( + ) C 0 = ( + ) + D = 0 E ( + 3) = 0. ANALYZING RELATIONSHIPS Which graph has -intercepts that are equivalent to the roots of the equation ( 3 ) = 5? Eplain our reasoning. A C 3 (, 0) B (1, 0) (, 0) (, 0) ERROR ANALYSIS In Eercises 3 and, describe and correct the error in solving the equation. 3.. D ( + 1) + 3 = 1 ( + 1) = 18 ( + 1) = = 3 8 = 0 = 8 = = = ± ( 1, 0) (, 0) (, 0) Section.1 Solving Quadratic Equations 151

11 5. OPEN-ENDED Write an equation of the form = d that has (a) two real solutions, (b) one real solution, and (c) no real solution.. ANALYZING EQUATIONS Which equation has one real solution? Eplain. A 3 + = ( + 8) B 5 = C ( + 3) = 18 In Eercises 7 5, find the zero(s) of the function. (See Eample.) 7. g() = f () = h() = g() = f () = 1 5. f () = g() = h() = D 3 5 = REASONING Write a quadratic function in the form f () = + b + c that has zeros 8 and 11. In Eercises 7 3, solve the equation b factoring. (See Eample 3.) 7. 0 = = z 10z = = n n = 0 3. a 9 = w 1w = 1w = + 5. NUMBER SENSE Write a quadratic equation in standard form that has roots equidistant from 10 on the number line. 57. PROBLEM SOLVING A restaurant sells 330 sandwiches each da. For each \$0.5 decrease in price, the restaurant sells about 15 more sandwiches. How much should the restaurant charge to maimize dail revenue? What is the maimum dail revenue? (See Eample 5.) MATHEMATICAL CONNECTIONS In Eercises 35 38, find the value of. 35. Area of rectangle = 3 3. Area of circle = 5π Area of triangle = 38. Area of trapezoid = In Eercises 39, solve the equation using an method. Eplain our reasoning. 39. u = 9u 0. t = ( + 9) =. ( + ) = ( ) 18 = 10. t + 8t + 1 = = = PROBLEM SOLVING An athletic store sells about 00 pairs of basketball shoes per month when it charges \$10 per pair. For each \$ increase in price, the store sells two fewer pairs of shoes. How much should the store charge to maimize monthl revenue? What is the maimum monthl revenue? 59. MODELING WITH MATHEMATICS Niagara Falls is made up of three waterfalls. The height of the Canadian Horseshoe Falls is about 188 feet above the lower Niagara River. A log falls from the top of Horseshoe Falls. (See Eample.) a. Write a function that gives the height h (in feet) of the log after t seconds. How long does the log take to reach the river? b. Find and interpret h() h(3). 15 Chapter Quadratic Equations and Comple Numbers

12 0. MODELING WITH MATHEMATICS According to legend, in 1589, the Italian scientist Galileo Galilei dropped rocks of different weights from the top of the Leaning Tower of Pisa to prove his conjecture that the rocks would hit the ground at the same time. The height h (in feet) of a rock after t seconds can be modeled b h(t) = 19 1t.. CRITICAL THINKING Write and solve an equation to find two consecutive odd integers whose product is MATHEMATICAL CONNECTIONS A quadrilateral is divided into two right triangles as shown in the figure. What is the length of each side of the quadrilateral? ft a. Find and interpret the zeros of the function. Then use the zeros to sketch the graph. b. What do the domain and range of the function represent in this situation? 1. PROBLEM SOLVING You make a rectangular quilt that is 5 feet b feet. You use the remaining 10 square feet of fabric to add a border of uniform width to the quilt. What is the width of the border? +. ABSTRACT REASONING Suppose the equation a + b + c = 0 has no real solution and a graph of the related function has a verte that lies in the second quadrant. a. Is the value of a positive or negative? Eplain our reasoning. b. Suppose the graph is translated so the verte is in the fourth quadrant. Does the graph have an -intercepts? Eplain. 7. REASONING When an object is dropped on an planet, its height h (in feet) after t seconds can be modeled b the function h = g t + h 0, where h 0 is the object s initial height and g is the planet s acceleration due to gravit. Suppose a rock is dropped from the same initial height on the three planets shown. Make a conjecture about which rock will hit the ground first. Justif our answer MODELING WITH MATHEMATICS You drop a seashell into the ocean from a height of 0 feet. Write an equation that models the height h (in feet) of the seashell above the water after t seconds. How long is the seashell in the air? 3. WRITING The equation h = 0.019s models the height h (in feet) of the largest ocean waves when the wind speed is s knots. Compare the wind speeds required to generate 5-foot waves and 0-foot waves. Earth: g = 3 ft/sec Mars: g = 1 ft/sec Jupiter: g = 7 ft/sec 8. PROBLEM SOLVING A café has an outdoor, rectangular patio. The owner wants to add 39 square feet to the area of the patio b epanding the eisting patio as shown. Write and solve an equation to find the value of. B what distance should the patio be etended? Before After 5 ft 5 ft 15 ft 15 ft Section.1 Solving Quadratic Equations 153

13 9. PROBLEM SOLVING A flea can jump ver long distances. The path of the jump of a flea can be modeled b the graph of the function = , where is the horizontal distance (in inches) and is the vertical distance (in inches). Graph the function. Identif the verte and zeros and interpret their meanings in this situation. 70. HOW DO YOU SEE IT? An artist is painting a mural and drops a paintbrush. The graph represents the height h (in feet) of the paintbrush after t seconds. Height (feet) Height of Dropped Paintbrush Time (seconds) 73. DRAWING CONCLUSIONS Is there a formula for factoring the sum of two squares? You will investigate this question in parts (a) and (b). a. Consider the sum of squares + 9. If this sum can be factored, then there are integers m and n such that + 9 = ( + m)( + n). Write two equations that m and n must satisf. b. Show that there are no integers m and n that satisf both equations ou wrote in part (a). What can ou conclude? 7. THOUGHT PROVOKING You are redesigning a rectangular raft. The raft is feet long and feet wide. You want to double the area of the raft b adding to the eisting design. Draw a diagram of the new raft. Write and solve an equation ou can use to find the dimensions of the new raft. ft a. What is the initial height of the paintbrush? b. How long does it take the paintbrush to reach the ground? Eplain. 71. MAKING AN ARGUMENT Your friend claims the equation + 7 =9 can be solved b factoring and has a solution of = 7. You solve the equation b graphing the related function and claim there is no solution. Who is correct? Eplain. 7. ABSTRACT REASONING Factor the epressions and 9. Recall that an epression in this form is called a difference of two squares. Use our answers to factor the epression a. Graph the related function = a. Label the verte, -intercepts, and ais of smmetr. Maintaining Mathematical Proficienc Find the sum or difference. (Skills Review Handbook) ft 75. MODELING WITH MATHEMATICS A high school wants to double the size of its parking lot b epanding the eisting lot as shown. B what distance should the lot be epanded? 75 ft 15 ft 7. ( + ) + ( ) 77. ( 3 + ) + (3 + 10) epanded part of lot old lot school 300 ft 78. ( + 1) ( 3 + ) 79. ( ) ( + 3 9) Find the product. (Skills Review Handbook) 80. ( + )( ) 81. (3 + 5 ) 8. (7 )( 1) ( ) 75 ft Reviewing what ou learned in previous grades and lessons 15 Chapter Quadratic Equations and Comple Numbers

14 . TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..F A.7.A Comple Numbers Essential Question What are the subsets of the set of comple numbers? In our stud of mathematics, ou have probabl worked with onl real numbers, which can be represented graphicall on the real number line. In this lesson, the sstem of numbers is epanded to include imaginar numbers. The real numbers and imaginar numbers compose the set of comple numbers. Comple Numbers Real Numbers Imaginar Numbers Rational Numbers Irrational Numbers Integers Whole Numbers The imaginar unit i is defined as i = 1. Natural Numbers Classifing Numbers USING PRECISE MATHEMATICAL LANGUAGE To be proficient in math, ou need to use clear definitions in our reasoning and discussions with others. Work with a partner. Determine which subsets of the set of comple numbers contain each number. a. 9 b. 0 c. d. 9 e. f. 1 Comple Solutions of Quadratic Equations Work with a partner. Use the definition of the imaginar unit i to match each quadratic equation with its comple solution. Justif our answers. a. = 0 b. + 1 = 0 c. 1 = 0 d. + = 0 e. 9 = 0 f. + 9 = 0 A. i B. 3i C. 3 D. i E. 1 F. Communicate Your Answer 3. What are the subsets of the set of comple numbers? Give an eample of a number in each subset.. Is it possible for a number to be both whole and natural? natural and rational? rational and irrational? real and imaginar? Eplain our reasoning. Section. Comple Numbers 155

15 . Lesson What You Will Learn Core Vocabular imaginar unit i, p. 15 comple number, p. 15 imaginar number, p. 15 pure imaginar number, p. 15 Define and use the imaginar unit i. Add, subtract, and multipl comple numbers. Find comple solutions and zeros. The Imaginar Unit i Not all quadratic equations have real-number solutions. For eample, = 3 has no real-number solutions because the square of an real number is never a negative number. To overcome this problem, mathematicians created an epanded sstem of numbers using the imaginar unit i, defined as i = 1. Note that i = 1. The imaginar unit i can be used to write the square root of an negative number. Core Concept The Square Root of a Negative Number Propert Eample 1. If r is a positive real number, then r = i r. 3 = i 3. B the first propert, it follows that ( i r ) = r. ( i 3 ) = i 3 = 3 Finding Square Roots of Negative Numbers Find the square root of each number. a. 5 b. 7 c. 5 9 a. 5 = 5 1 = 5i b. 7 = 7 1 = 3 i = i = i c. 5 9 = = 5 3 i = 15i Monitoring Progress Find the square root of the number. Help in English and Spanish at BigIdeasMath.com A comple number written in standard form is a number a + bi where a and b are real numbers. The number a is the real part, and the number bi is the imaginar part. a + bi If b 0, then a + bi is an imaginar number. If a = 0 and b 0, then a + bi is a pure imaginar number. The diagram shows how different tpes of comple numbers are related. Comple Numbers (a + bi) Real Numbers (a + 0i) π Imaginar Numbers (a + bi, b 0) + 3i 9 5i Pure Imaginar Numbers (0 + bi, b 0) i i 15 Chapter Quadratic Equations and Comple Numbers

16 Two comple numbers a + bi and c + di are equal if and onl if a = c and b = d. Equalit of Two Comple Numbers Find the values of and that satisf the equation 7i = 10 + i. Set the real parts equal to each other and the imaginar parts equal to each other. = 10 Equate the real parts. 7i = i Equate the imaginar parts. = 5 Solve for. 7 = Solve for. So, = 5 and = 7. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the values of and that satisf the equation i = 9 i. 9 + i = + 3i Operations with Comple Numbers Core Concept Sums and Differences of Comple Numbers To add (or subtract) two comple numbers, add (or subtract) their real parts and their imaginar parts separatel. Sum of comple numbers: Difference of comple numbers: (a + bi) + (c + di) = (a + c) + (b + d)i (a + bi) (c + di) = (a c) + (b d)i Adding and Subtracting Comple Numbers Add or subtract. Write the answer in standard form. a. (8 i ) + (5 + i ) b. (7 i) (3 i) c. 13 ( + 7i) + 5i a. (8 i ) + (5 + i ) = (8 + 5) + ( 1 + )i Definition of comple addition = i Write in standard form. b. (7 i ) (3 i ) = (7 3) + ( + )i Definition of comple subtraction = + 0i Simplif. = Write in standard form. c. 13 ( + 7i ) + 5i = [(13 ) 7i] + 5i Definition of comple subtraction = (11 7i ) + 5i Simplif. = 11 + ( 7 + 5)i Definition of comple addition = 11 i Write in standard form. Section. Comple Numbers 157

17 Solving a Real-Life Problem Electrical circuit components, such as resistors, inductors, and capacitors, all oppose the flow of current. This opposition is called resistance for resistors and reactance for inductors and capacitors. Each of these quantities is measured in ohms. The smbol used for ohms is Ω, the uppercase Greek letter omega. Component and smbol Resistor Inductor Capacitor 5Ω Resistance or reactance (in ohms) R L C 3Ω Ω Impedance (in ohms) R Li Ci Alternating current source The table shows the relationship between a component s resistance or reactance and its contribution to impedance. A series circuit is also shown with the resistance or reactance of each component labeled. The impedance for a series circuit is the sum of the impedances for the individual components. Find the impedance of the circuit. The resistor has a resistance of 5 ohms, so its impedance is 5 ohms. The inductor has a reactance of 3 ohms, so its impedance is 3i ohms. The capacitor has a reactance of ohms, so its impedance is i ohms. Impedance of circuit = 5 + 3i + (i) = 5 i The impedance of the circuit is (5 i ) ohms. To multipl two comple numbers, use the Distributive Propert, or the FOIL method, just as ou do when multipling real numbers or algebraic epressions. Multipling Comple Numbers STUDY TIP When simplifing an epression that involves comple numbers, be sure to simplif i as 1. Multipl. Write the answer in standard form. a. i( + i ) b. (9 i )( + 7i ) a. i( + i ) = i + i Distributive Propert = i + ( 1) Use i = 1. = i Write in standard form. b. (9 i )( + 7i ) = 3 + 3i + 8i 1i Multipl using FOIL. = i 1( 1) Simplif and use i = 1. = i + 1 Simplif. = + 71i Write in standard form. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 7. WHAT IF? In Eample, what is the impedance of the circuit when the capacitor is replaced with one having a reactance of 7 ohms? Perform the operation. Write the answer in standard form. 8. (9 i ) + ( + 7i ) 9. (3 + 7i ) (8 i ) 10. (1 + i ) (5 + 9i ) 11. ( 3i)(10i) 1. i(8 i ) 13. (3 + i )(5 i ) 158 Chapter Quadratic Equations and Comple Numbers

18 Comple Solutions and Zeros Solving Quadratic Equations ANALYZING MATHEMATICAL RELATIONSHIPS Notice that ou can use the solutions in Eample (a) to factor + as ( + i )( i ). Solve (a) + = 0 and (b) 11 = 7. a. + = 0 Write original equation. = Subtract from each side. = ± Take square root of each side. = ±i Write in terms of i. The solutions are i and i. b. 11 = 7 Write original equation. = 3 Add 11 to each side. = 18 Divide each side b. = ± 18 Take square root of each side. = ±i 18 Write in terms of i. = ±3i Simplif radical. The solutions are 3i and 3i. Finding Zeros of a Quadratic Function FORMULATING A PLAN The graph of f does not intersect the -ais, which means f has no real zeros. So, f must have comple zeros, which ou can find algebraicall. 30 f() = Find the zeros of f () = = 0 Set f () equal to 0. = 0 Subtract 0 from each side. = 5 Divide each side b. = ± 5 Take square root of each side. = ±i 5 Write in terms of i. So, the zeros of f are i 5 and i 5. Check f ( i 5 ) = ( i 5 ) + 0 = 5i + 0 = ( 5) + 0 = 0 f ( i 5 ) = ( i 5 ) + 0 = 5i + 0 = ( 5) + 0 = 0 Monitoring Progress Solve the equation. Help in English and Spanish at BigIdeasMath.com 1. = = = = = = 3 Find the zeros of the function. 0. f () = f () =. f () = Section. Comple Numbers 159

19 . Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check 1. VOCABULARY What is the imaginar unit i defined as and how can ou use i?. COMPLETE THE SENTENCE For the comple number 5 + i, the imaginar part is and the real part is. 3. WRITING Describe how to add comple numbers.. WHICH ONE DOESN T BELONG? Which number does not belong with the other three? Eplain our reasoning i + 5i 3 + i 0 7i Monitoring Progress and Modeling with Mathematics In Eercises 5 1, find the square root of the number. (See Eample 1.) In Eercises 13 0, find the values of and that satisf the equation. (See Eample.) i = 8 + i i = 7 + i i = 0 + 3i i = 3 + i 17. i = 1 + 1i 3. (1 + i ) (3 7i ). ( 15i ) ( + 5i ) 5. (1 3i ) + (7 + 3i ). (1 9i ) ( 9i ) 7. 7 (3 + i ) + i 8. 1 ( 3i ) i ( 5i ) 9i (8 + i ) + 7i 31. USING STRUCTURE Write each epression as a comple number in standard form. a b REASONING The additive inverse of a comple number z is a comple number z a such that z + z a = 0. Find the additive inverse of each comple number. a. z = 1 + i b. z = 3 i c. z = + 8i In Eercises 33 3, find the impedance of the series circuit. (See Eample.) i = 0 13i 33. 1Ω 3. Ω i = 9 i 7 9Ω 7Ω Ω 9Ω i = 1 + i In Eercises 1 30, add or subtract. Write the answer in standard form. (See Eample 3.) 35. 8Ω 3Ω 3. 1Ω 7Ω 1. ( i ) + (7 + 3i ). (9 + 5i ) + (11 + i ) Ω 8Ω 10 Chapter Quadratic Equations and Comple Numbers

20 In Eercises 37, multipl. Write the answer in standard form. (See Eample 5.) 37. 3i( 5 + i ) 38. i(7 i ) 39. (3 i )( + i ) 0. (7 + 5i )(8 i ) 1. ( i )( + i ). (9 + 5i )(9 5i ) 3. (3 i ). (8 + 3i ) JUSTIFYING STEPS In Eercises 5 and, justif each step in performing the operation ( + 3i ) + 5i = [(11 ) 3i ] + 5i = (7 3i ) + 5i = 7 + ( 3 + 5)i = 7 + i. (3 + i )(7 i ) = 1 1i + 1i 8i = 1 + i 8( 1) = 1 + i + 8 = 9 + i REASONING In Eercises 7 and 8, place the tiles in the epression to make a true statement. 7. ( i ) ( i ) = i i( + i ) = 18 10i 5 9 In Eercises 9 5, solve the equation. Check our solution(s). (See Eample.) = = 0 In Eercises 55, find the zeros of the function. (See Eample 7.) 55. f () = g() = h() = k() = m() = 7 0. p() = r () =. f () = 5 10 ERROR ANALYSIS In Eercises 3 and, describe and correct the error in performing the operation and writing the answer in standard form NUMBER SENSE Simplif each epression. Then classif our results in the table below. a. ( + 7i ) + ( 7i ) b. ( i ) ( 10 + i ) c. (5 + 15i ) (5 i ) d. (5 + i )(8 i ) e. (17 3i ) + ( 17 i ) f. ( 1 + i )(11 i ) g. (7 + 5i ) + (7 5i ) h. ( 3 + i ) ( 3 8i ) Real numbers (3 + i )(5 i ) = 15 3i + 10i i Imaginar numbers = i i = i + 7i + 15 ( + i ) = () + (i ) 1 = 1 + 3i = 1 + (3)( 1) = 0 Pure imaginar numbers 51. = = = = 7. MAKING AN ARGUMENT The Product Propert states a b = ab. Your friend concludes 9 = 3 =. Is our friend correct? Eplain. Section. Comple Numbers 11

21 7. FINDING A PATTERN Make a table that shows the powers of i from i 1 to i 8 in the first row and the simplified forms of these powers in the second row. Describe the pattern ou observe in the table. Verif the pattern continues b evaluating the net four powers of i. 8. HOW DO YOU SEE IT? The graphs of three functions are shown. Which function(s) has real zeros? imaginar zeros? Eplain our reasoning. h f g In Eercises 9 7, write the epression as a comple number in standard form. 9. (3 + i ) (7 5i ) + i(9 + 1i ) 70. 3i( + 5i ) + ( 7i ) (9 + i ) 71. (3 + 5i )( 7i ) 7. i 3 (5 1i ) 73. ( + i 5 ) + (1 9i ) ( 3 + i 7 ) 7. (8 i ) + (3 7i 8 ) ( + i 9 ) 75. OPEN-ENDED Find two imaginar numbers whose sum and product are real numbers. How are the imaginar numbers related? 7. COMPARING METHODS Describe the two different methods shown for writing the comple epression in standard form. Which method do ou prefer? Eplain. Method 1 i ( 3i ) + i (1 i ) = 8i 1i + i 8i = 8i 1( 1) + i 8( 1) = 0 + 1i Method i( 3i ) + i (1 i ) = i [( 3i ) + (1 i )] = i [3 5i ] = 1i 0i = 1i 0( 1) = 0 + 1i 77. CRITICAL THINKING Determine whether each statement is true or false. If it is true, give an eample. If it is false, give a countereample. a. The sum of two imaginar numbers is an imaginar number. b. The product of two pure imaginar numbers is a real number. c. A pure imaginar number is an imaginar number. d. A comple number is a real number. 78. THOUGHT PROVOKING Create a circuit that has an impedance of 1 3i. Maintaining Mathematical Proficienc Determine whether the given value of is a solution to the equation. Reviewing what ou learned in previous grades and lessons (Skills Review Handbook) 79. 3( ) + 1 = 1; = = + 9 3; = = 19 3 ; = Write an equation in verte form of the parabola whose graph is shown. (Section 3.) 3 8. (0, 3) 83. ( 1, 5) 8. (3, ) (1, ) (, 1) ( 3, 3) 1 Chapter Quadratic Equations and Comple Numbers

22 .3 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..D A..F Completing the Square Essential Question How can ou complete the square for a quadratic epression? Using Algebra Tiles to Complete the Square Work with a partner. Use algebra tiles to complete the square for the epression +. a. You can model + using one -tile and si -tiles. Arrange the tiles in a square. Your arrangement will be incomplete in one of the corners. b. How man 1-tiles do ou need to complete the square? c. Find the value of c so that the epression + + c is a perfect square trinomial. d. Write the epression in part (c) as the square of a binomial. Drawing Conclusions Work with a partner. a. Use the method outlined in Eploration 1 to complete the table. ANALYZING MATHEMATICAL RELATIONSHIPS To be proficient in math, ou need to look closel to discern a pattern or structure. Epression + + c + + c c c Value of c needed to complete the square Epression written as a binomial squared b. Look for patterns in the last column of the table. Consider the general statement + b + c = ( + d ). How are d and b related in each case? How are c and d related in each case? c. How can ou obtain the values in the second column directl from the coefficients of in the first column? Communicate Your Answer 3. How can ou complete the square for a quadratic epression?. Describe how ou can solve the quadratic equation + = 1 b completing the square. Section.3 Completing the Square 13

24 Solving Quadratic Equations b Completing the Square Making a Perfect Square Trinomial Find the value of c that makes c a perfect square trinomial. Then write the epression as the square of a binomial. 1 Step 1 Find half the coefficient of. = 7 Step Square the result of Step 1. 7 = Step 3 Replace c with the result of Step The epression c is a perfect square trinomial when c = 9. Then = ( + 7)( + 7) = ( + 7). Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the value of c that makes the epression a perfect square trinomial. Then write the epression as the square of a binomial c 5. + c. 9 + c ANALYZING MATHEMATICAL RELATIONSHIPS Notice ou cannot solve the equation b factoring because is not factorable as a product of binomials. The method of completing the square can be used to solve an quadratic equation. When ou complete the square as part of solving an equation, ou must add the same number to both sides of the equation. Solving a + b + c = 0 when a = 1 Solve = 0 b completing the square = 0 Write the equation. 10 = 7 Write left side in the form + b = Add ( b ) = ( 10 ) = 5 to each side. ( 5) = 18 Write left side as a binomial squared. 5 = ± 18 = 5 ± 18 = 5 ± 3 Take square root of each side. Add 5 to each side. Simplif radical. The solutions are = and = 5 3. You can check this b graphing = The -intercepts are about and Check 8 1 Zero X=9.07 Y=0 Section.3 Completing the Square 15

25 Solving a + b + c = 0 when a 1 Solve = 0 b completing the square. The coefficient a is not 1, so ou must first divide each side of the equation b a = 0 Write the equation = 0 Divide each side b 3. + = 5 Write left side in the form + b. + + = 5 + Add ( b ) = ( ) = to each side. ( + ) = 1 Write left side as a binomial squared. + = ± 1 = ± 1 Take square root of each side. Subtract from each side. = ± i Write in terms of i. The solutions are = + i and = i. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Solve the equation b completing the square = = = = ( + ) = 1. ( ) = 00 Writing Quadratic Functions in Verte Form Recall that the verte form of a quadratic function is = a( h) + k, where (h, k) is the verte of the graph of the function. You can write a quadratic function in verte form b completing the square. Writing a Quadratic Function in Verte Form Check 1 Minimum X= Y=-18 1 Write = in verte form. Then identif the verte. = ? = ( 1 +?) = ( 1 + 3) + 18 Add ( b ) + 3 = ( ) + 18 Write the function. = ( ) 18 Solve for. Prepare to complete the square. = ( 1 ) = 3 to each side. Write as a binomial squared. The verte form of the function is = ( ) 18. The verte is (, 18). Monitoring Progress Help in English and Spanish at BigIdeasMath.com Write the quadratic function in verte form. Then identif the verte. 13. = = = 1 Chapter Quadratic Equations and Comple Numbers

26 Modeling with Mathematics The height (in feet) of a baseball t seconds after it is hit can be modeled b the function = 1t + 9t + 3. Find the maimum height of the baseball. How long does the ball take to hit the ground? ANOTHER WAY You can use the coefficients of the original function = f () to find the maimum height. b f ( a) ( = f 9 ( 1)) = f (3) = 17 ANALYZING MATHEMATICAL RELATIONSHIPS You could write the zeros as 3 ± 7 3, but it is easier to recognize that is negative because is greater than Understand the Problem You are given a quadratic function that represents the height of a ball. You are asked to determine the maimum height of the ball and how long it is in the air.. Make a Plan Write the function in verte form to identif the maimum height. Then find and interpret the zeros to determine how long the ball takes to hit the ground. 3. Solve the Problem Write the function in verte form b completing the square. = 1t + 9t + 3 Write the function. = 1(t t) + 3 +? = 1(t t +?) ( 1)(9) = 1(t t + 9) = 1(t 3) + 3 = 1(t 3) + 17 Solve for. Factor 1 from first two terms. Prepare to complete the square. Add ( 1)(9) to each side. The verte is (3, 17). Find the zeros of the function. 0 = 1(t 3) + 17 Substitute 0 for. 17 = 1(t 3) Write t t + 9 as a binomial squared. Subtract 17 from each side = (t 3) Divide each side b 1. ± = t 3 Take square root of each side. 3 ± = t Add 3 to each side. Reject the negative solution, , because time must be positive. So, the maimum height of the ball is 17 feet, and it takes seconds for the ball to hit the ground.. Look Back The verte indicates that the maimum height of 17 feet occurs when t = 3. This makes sense because the graph of the function is parabolic with zeros near t = 0 and t =. You can use a graph to check the maimum height. Monitoring Progress Maimum 0 X=3 Y=17 0 Help in English and Spanish at BigIdeasMath.com 1. WHAT IF? The height of the baseball can be modeled b = 1t + 80t +. Find the maimum height of the baseball. How long does the ball take to hit the ground? Section.3 Completing the Square 17

27 .3 Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check 1. VOCABULARY What must ou add to the epression + b to complete the square?. COMPLETE THE SENTENCE The trinomial + 9 is a because it equals. Monitoring Progress and Modeling with Mathematics In Eercises 3 10, solve the equation using square roots. Check our solution(s). (See Eample 1.) = 5. r 10r + 5 = = 5. m + 8m + 1 = = = w + w + 1 = = 1 In Eercises 11 0, find the value of c that makes the epression a perfect square trinomial. Then write the epression as the square of a binomial. (See Eample.) c c c 1. t t + c c c 17. z 5z + c c 19. w + 13w + c 0. s s + c In Eercises 1, find the value of c. Then write an epression represented b the diagram c c In Eercises 5 3, solve the equation b completing the square. (See Eamples 3 and.) = 0. s + s = = 0 8. t 8t 5 = 0 9. z(z + 9) = ( + 8) = t + 8t + 5 = 0 3. r + r + 1 = ( + ) = w(w 3) = = s + 8s = s ERROR ANALYSIS Describe and correct the error in solving the equation = 0 ( + ) = 11 ( ) = ( + 3) = 0 ( + 3) = = ± 5 = 3 ± ERROR ANALYSIS Describe and correct the error in finding the value of c that makes the epression a perfect square trinomial c c c 39. WRITING Can ou solve an equation b completing the square when the equation has two imaginar solutions? Eplain. 18 Chapter Quadratic Equations and Comple Numbers

28 0. ABSTRACT REASONING Which of the following are solutions of the equation a + a = b? Justif our answers. A ab B a b C b D a E a b F a + b USING STRUCTURE In Eercises 1 50, determine whether ou would use factoring, square roots, or completing the square to solve the equation. Eplain our reasoning. Then solve the equation = = 0 3. ( + ) = 1. ( 7) = = = = = 0 0. f() = f() = 3 +. g() = MODELING WITH MATHEMATICS While marching, a drum major tosses a baton into the air and catches it. The height h (in feet) of the baton t seconds after it is thrown can be modeled b the function h = 1t + 3t +. (See Eample.) a. Find the maimum height of the baton. b. The drum major catches the baton when it is feet above the ground. How long is the baton in the air?. MODELING WITH MATHEMATICS A firework eplodes when it reaches its maimum height. The height h (in feet) of the firework t seconds after it is launched can be modeled b h = t t What is the maimum height of the firework? How long is the firework in the air before it eplodes? = = 0 MATHEMATICAL CONNECTIONS In Eercises 51 5, find the value of. 51. Area of 5. Area of rectangle = 50 parallelogram = Area of triangle = 0 5. Area of trapezoid = In Eercises 55, write the quadratic function in verte form. Then identif the verte. (See Eample 5.) 55. f() = g() = g() = h() = h() = COMPARING METHODS A skateboard shop sells about 50 skateboards per week when the advertised price is charged. For each \$1 decrease in price, one additional skateboard per week is sold. The shop s revenue can be modeled b = (70 )(50 + ). SKATEBOARDS Qualit Skateboards for \$70 a. Use the intercept form of the function to find the maimum weekl revenue. b. Write the function in verte form to find the maimum weekl revenue. c. Which wa do ou prefer? Eplain our reasoning. Section.3 Completing the Square 19

29 . HOW DO YOU SEE IT? The graph of the function f () = ( h) is shown. What is the -intercept? Eplain our reasoning. f (0, 9) 9. MAKING AN ARGUMENT Your friend sas the equation + 10 = 0 can be solved b either completing the square or factoring. Is our friend correct? Eplain. 70. THOUGHT PROVOKING Write a function g in standard form whose graph has the same -intercepts as the graph of f () = Find the zeros of each function b completing the square. Graph each function. 71. CRITICAL THINKING Solve + b + c = 0 b completing the square. Your answer will be an epression for in terms of b and c. 7. WRITING At Buckingham Fountain in Chicago, the height h (in feet) of the water above the main nozzle can be modeled b h = 1t + 89.t, where t is the time (in seconds) since the water has left the nozzle. Describe three different was ou could find the maimum height the water reaches. Then choose a method and find the maimum height of the water. 8. PROBLEM SOLVING A farmer is building a rectangular pen along the side of a barn for animals. The barn will serve as one side of the pen. The farmer has 10 feet of fence to enclose an area of 151 square feet and wants each side of the pen to be at least 0 feet long. a. Write an equation that represents the area of the pen. b. Solve the equation in part (a) to find the dimensions of the pen. 7. DRAWING CONCLUSIONS In this eercise, ou will investigate the graphical effect of completing the square. a. Graph each pair of functions in the same coordinate plane. = + = = ( + 1) = ( 3) b. Compare the graphs of = + b and = ( + b ). Describe what happens to the graph of = + b when ou complete the square. 73. MODELING WITH MATHEMATICS In our potter class, ou are given a lump of cla with a volume of 00 cubic centimeters and are asked to make a clindrical pencil holder. The pencil holder should be 9 centimeters high and have an inner radius of 3 centimeters. What thickness should our pencil holder have if ou want to use all of the cla? Top view 3 cm cm 3 cm cm 9 cm cm cm Side view Maintaining Mathematical Proficienc Solve the inequalit. Graph the solution. (Skills Review Handbook) n s 7. 3 < > Graph the function. Label the verte, ais of smmetr, and -intercepts. (Section 3.) 78. g() = ( ) 79. h() = ( 3) 80. f () = f () = ( + 10)( 1) Reviewing what ou learned in previous grades and lessons 170 Chapter Quadratic Equations and Comple Numbers

30 .1.3 What Did You Learn? Core Vocabular quadratic equation in one variable, p. 1 root of an equation, p. 1 zero of a function, p. 18 imaginar unit i, p. 15 comple number, p. 15 imaginar number, p. 15 pure imaginar number, p. 15 completing the square, p. 1 Core Concepts Section.1 Solving Quadratic Equations Graphicall, p. 1 Solving Quadratic Equations Algebraicall, p. 17 Zero-Product Propert, p. 18 Section. The Square Root of a Negative Number, p. 15 Operations with Comple Numbers, p. 157 Section.3 Solving Quadratic Equations b Completing the Square, p. 15 Writing Quadratic Functions in Verte Form, p. 1 Mathematical Thinking 1. Analze the givens, constraints, relationships, and goals in Eercise 1 on page Determine whether it would be easier to find the zeros of the function in Eercise 3 on page 19 or Eercise 7 on page 170. Stud Skills Creating a Positive Stud Environment Set aside an appropriate amount of time for reviewing our notes and the tetbook, reworking our notes, and completing homework. Set up a place for studing at home that is comfortable, but not too comfortable. The place needs to be awa from all potential distractions. Form a stud group. Choose students who stud well together, help out when someone misses school, and encourage positive attitudes. 171

31 .1.3 Quiz Solve the equation b using the graph. Check our solution(s). (Section.1) = = 1 3. = f() = g() = h() = + 8 Solve the equation using square roots or b factoring. Eplain the reason for our choice. (Section.1). 15 = = 0. ( + 3) = 8 7. Find the values of and that satisf the equation 7 i = 1 + i. (Section.) Perform the operation. Write our answer in standard form. (Section.) 8. ( + 5i ) + ( + 3i ) 9. (3 + 9i ) (1 7i ) 10. ( + i )( 3 5i ) 11. Find the zeros of the function f () = 9 +. Does the graph of the function intersect the -ais? Eplain our reasoning. (Section.) Solve the equation b completing the square. (Section.3) = = 0 1. ( + ) = Write = 10 + in verte form. Then identif the verte. (Section.3) 1. A museum has a café with a rectangular patio. The museum wants to add square feet to the area of the patio b epanding the eisting patio as shown. (Section.1) a. Find the area of the eisting patio. b. Write an equation to model the area of the new patio. c. B what distance should the length of the patio be epanded? 17. Find the impedance of the series circuit. (Section.) Eisting patio 30 ft 0 ft 7Ω 5Ω Ω 18. The height h (in feet) of a badminton birdie t seconds after it is hit can be modeled b the function h = 1t + 3t +. (Section.3) a. Find the maimum height of the birdie. b. How long is the birdie in the air? 17 Chapter Quadratic Equations and Comple Numbers

34 ANOTHER WAY You can also use factoring to solve = 0 because the left side factors as (5 ). Check Solving an Equation with One Real Solution Solve 5 8 = 1 using the Quadratic Formula. 5 8 = = 0 = ( 0) ± ( 0) (5)() (5) = 0 ± 0 50 Write original equation. Write in standard form. a = 5, b = 0, c = Simplif. 0.5 Zero X=. Y= = 5 Simplif. So, the solution is = 5. You can check this b graphing = The onl -intercept is 5. Solving an Equation with Imaginar Solutions Solve + = 13 using the Quadratic Formula. COMMON ERROR Remember to divide the real part and the imaginar part b when simplifing. + = = 0 = ± ( 1)( 13) ( 1) = ± 3 ± i = = ± 3i The solutions are = + 3i and = 3i. Write original equation. Write in standard form. a = 1, b =, c = 13 Simplif. Write in terms of i. Simplif. Check Graph = There are no -intercepts. So, the original equation has no real solutions. The algebraic check for one of the imaginar solutions is shown ( + 3i ) + ( + 3i ) =? i i =? = Monitoring Progress Solve the equation using the Quadratic Formula. Help in English and Spanish at BigIdeasMath.com. + 1 = = = 3 + Section. Using the Quadratic Formula 175

35 Analzing the Discriminant In the Quadratic Formula, the epression b ac is called the discriminant of the associated equation a + b + c = 0. = b ± b ac a discriminant You can analze the discriminant of a quadratic equation to determine the number and tpe of solutions of the equation. Core Concept Analzing the Discriminant of a + b + c = 0 Value of discriminant b ac > 0 b ac = 0 b ac < 0 Number and tpe of solutions Two real solutions One real solution Two imaginar solutions Graph of = a + b + c Two -intercepts One -intercept No -intercept Analzing the Discriminant Find the discriminant of the quadratic equation and describe the number and tpe of solutions of the equation. a = 0 b. + 9 = 0 c. + 8 = 0 Equation Discriminant Solution(s) a + b + c = 0 b ac = b ± b ac a a = 0 ( ) (1)(10) = Two imaginar: 3 ± i b. + 9 = 0 ( ) (1)(9) = 0 One real: 3 c. + 8 = 0 ( ) (1)(8) = Two real:, Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the discriminant of the quadratic equation and describe the number and tpe of solutions of the equation = = = = = = Chapter Quadratic Equations and Comple Numbers

36 Writing an Equation Find a possible pair of integer values for a and c so that the equation a + c = 0 has one real solution. Then write the equation. In order for the equation to have one real solution, the discriminant must equal 0. b ac = 0 Write the discriminant. ANOTHER WAY Another possible equation in Eample 5 is 1 = 0. You can obtain this equation b letting a = and c = 1. () ac = 0 Substitute for b. 1 ac = 0 Evaluate the power. ac = 1 Subtract 1 from each side. ac = Divide each side b. Because ac =, choose two integers whose product is, such as a = 1 and c =. So, one possible equation is + = 0. Check Graph = +. The onl -intercept is. You can also check b factoring. 8 + = 0 ( ) = 0 = 3 Zero X= Y=0 7 Monitoring Progress Help in English and Spanish at BigIdeasMath.com 13. Find a possible pair of integer values for a and c so that the equation a c = 0 has two real solutions. Then write the equation. The table shows five methods for solving quadratic equations. For a given equation, it ma be more efficient to use one method instead of another. Suggestions about when to use each method are shown below. Concept Summar Methods for Solving Quadratic Equations Method Graphing Using square roots Factoring Completing the square Quadratic Formula When to Use Use when approimate solutions are adequate. Use when solving an equation that can be written in the form u = d, where u is an algebraic epression. Use when a quadratic equation can be factored easil. Can be used for an quadratic equation a + b + c = 0 but is simplest to appl when a = 1 and b is an even number. Can be used for an quadratic equation. Section. Using the Quadratic Formula 177

37 Solving Real-Life Problems The function h = 1t + h 0 is used to model the height of a dropped object. For an object that is launched or thrown, an etra term v 0 t must be added to the model to account for the object s initial vertical velocit v 0 (in feet per second). Recall that h is the height (in feet), t is the time in motion (in seconds), and h 0 is the initial height (in feet). h = 1t + h 0 h = 1t + v 0 t + h 0 Object is dropped. Object is launched or thrown. As shown below, the value of v 0 can be positive, negative, or zero depending on whether the object is launched upward, downward, or parallel to the ground. V 0 > 0 V 0 < 0 V 0 = 0 Modeling a Launched Object A juggler tosses a ball into the air. The ball leaves the juggler s hand feet above the ground and has an initial vertical velocit of 30 feet per second. The juggler catches the ball when it falls back to a height of 3 feet. How long is the ball in the air? Because the ball is thrown, use the model h = 1t + v 0 t + h 0. To find how long the ball is in the air, solve for t when h = 3. h = 1t + v 0 t + h 0 Write the height model. 3 = 1t + 30t + Substitute 3 for h, 30 for v 0, and for h 0. 0 = 1t + 30t + 1 Write in standard form. This equation is not factorable, and completing the square would result in fractions. So, use the Quadratic Formula to solve the equation. t = 30 ± 30 ( 1)(1) a = 1, b = 30, c = 1 ( 1) t = 30 ± 9 3 t or t 1.9 Simplif. Use a calculator. Reject the negative solution, 0.033, because the ball s time in the air cannot be negative. So, the ball is in the air for about 1.9 seconds. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 1. WHAT IF? The ball leaves the juggler s hand with an initial vertical velocit of 0 feet per second. How long is the ball in the air? 178 Chapter Quadratic Equations and Comple Numbers

38 . Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE When a, b, and c are real numbers such that a 0, the solutions of the quadratic equation a + b + c = 0 are =.. COMPLETE THE SENTENCE You can use the of a quadratic equation to determine the number and tpe of solutions of the equation. 3. WRITING Describe the number and tpe of solutions when the value of the discriminant is negative.. WRITING Which two methods can ou use to solve an quadratic equation? Eplain when ou might prefer to use one method over the other. Monitoring Progress and Modeling with Mathematics In Eercises 5 18, solve the equation using the Quadratic Formula. Use a graphing calculator to check our solution(s). (See Eamples 1,, and 3.) = = = = = = = 1. 3 = = = = = 17. z = 1z w + = w In Eercises 19, find the discriminant of the quadratic equation and describe the number and tpe of solutions of the equation. (See Eample.) = = 0 1. n n = = 0 3. = p = p USING EQUATIONS Determine the number and tpe of solutions to the equation + 7 = 11. A B C D two real solutions one real solution two imaginar solutions one imaginar solution ANALYZING EQUATIONS In Eercises 9 3, use the discriminant to match each quadratic equation with the correct graph of the related function. Eplain our reasoning = = = = 0 A. 8 B = 8 3. = C. D USING EQUATIONS What are the comple solutions of the equation = 0? A + 3i, 3i B + 1i, 1i C 1 + 3i, 1 3i D 1 + 1i, 1 1i Section. Using the Quadratic Formula 179

39 ERROR ANALYSIS In Eercises 33 and 3, describe and correct the error in solving the equation = 0 = 10 ± 10 (1)(7) (1) = 10 ± ± 1 = = 1 or COMPARING METHODS In Eercises 7 58, solve the quadratic equation using the Quadratic Formula. Then solve the equation using another method. Which method do ou prefer? Eplain = = = = = = = = = = = = ± (1)(8) (1) = ± = ± = or 57. = = 0 MATHEMATICAL CONNECTIONS In Eercises 59 and 0, find the value for. 59. Area of the rectangle = m ( + ) m ( 9) m OPEN-ENDED In Eercises 35 0, find a possible pair of integer values for a and c so that the quadratic equation has the given solution(s). Then write the equation. (See Eample 5.) 35. a + + c = 0; two imaginar solutions 3. a + + c = 0; two real solutions 37. a 8 + c = 0; two real solutions 38. a + c = 0; one real solution 39. a + 10 = c; one real solution 0. + c = a ; two imaginar solutions 0. Area of the triangle = 8 ft (3 7) ft ( + 1) ft 1. MODELING WITH MATHEMATICS A lacrosse plaer throws a ball in the air from an initial height of 7 feet. The ball has an initial vertical velocit of 90 feet per second. Another plaer catches the ball when it is 3 feet above the ground. How long is the ball in the air? (See Eample.) USING STRUCTURE In Eercises 1, use the Quadratic Formula to write a quadratic equation that has the given solutions. 1. = 8 ± = ± = ±. = 15 ± 15. = 9 ± 137. = ±. NUMBER SENSE Suppose the quadratic equation a c = 0 has one real solution. Is it possible for a and c to be integers? rational numbers? Eplain our reasoning. Then describe the possible values of a and c. 180 Chapter Quadratic Equations and Comple Numbers

41 70. HOW DO YOU SEE IT? The graph of a quadratic function = a + b + c is shown. Determine whether each discriminant of a + b + c = 0 is positive, negative, or zero. Then state the number and tpe of solutions for each graph. Eplain our reasoning. a. b. 7. THOUGHT PROVOKING Describe a real-life stor that could be modeled b h = 1t + v 0 t + h 0. Write the height model for our stor and determine how long our object is in the air. 75. REASONING Show there is no quadratic equation a + b + c = 0 such that a, b, and c are real numbers and 3i and i are solutions. 7. MODELING WITH MATHEMATICS The Stratosphere Tower in Las Vegas is 91 feet tall and has a needle at its top that etends even higher into the air. A thrill ride called Big Shot catapults riders 10 feet up the needle and then lets them fall back to the launching pad. c. 71. CRITICAL THINKING Solve each absolute value equation. a. 3 1 = b. = + 7. MAKING AN ARGUMENT The class is asked to solve the equation = 0. You decide to solve the equation b completing the square. Your friend decides to use the Quadratic Formula. Whose method is more efficient? Eplain our reasoning. 73. ABSTRACT REASONING For a quadratic equation a + b + c = 0 with two real solutions, show b that the mean of the solutions is. How is this a fact related to the smmetr of the graph of = a + b + c? Maintaining Mathematical Proficienc Solve the sstem of linear equations b graphing. (Skills Review Handbook) = 78. = 1 + = = = 80. = + + = = 10 Graph the quadratic equation. Label the verte and ais of smmetr. (Section 3.) 81. = = = = 3 a. The height h (in feet) of a rider on the Big Shot can be modeled b h = 1t + v 0 t + 91, where t is the elapsed time (in seconds) after launch and v 0 is the initial velocit (in feet per second). Find v 0 using the fact that the maimum value of h is = 1081 feet. b. A brochure for the Big Shot states that the ride up the needle takes seconds. Compare this time to the time given b the model h = 1t + v 0 t + 91, where v 0 is the value ou found in part (a). Discuss the accurac of the model. Reviewing what ou learned in previous grades and lessons 18 Chapter Quadratic Equations and Comple Numbers

42 .5 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A.3.A A.3.C A.3.D Solving Nonlinear Sstems Essential Question How can ou solve a nonlinear sstem of equations? Solving Nonlinear Sstems of Equations Work with a partner. Match each sstem with its graph. Eplain our reasoning. Then solve each sstem using the graph. a. = b. = + c. = 5 = + = + = + 1 d. = + e. = + 1 f. = = + = + 1 = + + A. B C. D. 8 E. 5 F USING PROBLEM-SOLVING STRATEGIES To be proficient in math, ou need to plan a solution pathwa rather than simpl jumping into a solution attempt. 3 Solving Nonlinear Sstems of Equations Work with a partner. Look back at the nonlinear sstem in Eploration 1(f). Suppose ou want a more accurate wa to solve the sstem than using a graphical approach. a. Show how ou could use a numerical approach b creating a table. For instance, ou might use a spreadsheet to solve the sstem. b. Show how ou could use an analtical approach. For instance, ou might tr solving the sstem b substitution or elimination. 7 Communicate Your Answer 3. How can ou solve a nonlinear sstem of equations?. Would ou prefer to use a graphical, numerical, or analtical approach to solve the given nonlinear sstem of equations? Eplain our reasoning. = + 3 = + Section.5 Solving Nonlinear Sstems 183

43 .5 Lesson What You Will Learn Core Vocabular sstem of nonlinear equations, p. 18 Previous sstem of linear equations circle Solve sstems of nonlinear equations. Solve quadratic equations b graphing. Sstems of Nonlinear Equations Previousl, ou solved sstems of linear equations b graphing, substitution, and elimination. You can also use these methods to solve a sstem of nonlinear equations. In a sstem of nonlinear equations, at least one of the equations is nonlinear. For instance, the nonlinear sstem shown has a quadratic equation and a linear equation. = + = + 5 Equation 1 is nonlinear. Equation is linear. When the graphs of the equations in a sstem are a line and a parabola, the graphs can intersect in zero, one, or two points. So, the sstem can have zero, one, or two solutions, as shown. No solution One solution Two solutions When the graphs of the equations in a sstem are a parabola that opens up and a parabola that opens down, the graphs can intersect in zero, one, or two points. So, the sstem can have zero, one, or two solutions, as shown. No solution One solution Two solutions Solving a Nonlinear Sstem b Graphing Solve the sstem b graphing. = 1 Equation 1 = 1 Equation Graph each equation. Then estimate the point of intersection. The parabola and the line appear to intersect at the point (0, 1). Check the point b substituting the coordinates into each of the original equations. Equation 1 Equation = 1 = (0, 1) =? (0) (0) 1 1 =? (0) 1 1 = 1 1 = 1 The solution is (0, 1). 18 Chapter Quadratic Equations and Comple Numbers

44 Solving a Nonlinear Sstem b Substitution Height (feet) Soccer Kick (1, 1) 1 8 (, ) (0, 0) Time (seconds) Check 1 The graph shows the height (in feet) of a soccer ball seconds after it is kicked from the base of a hill with an initial velocit of 30 feet per second. The linear function represents a hill with a constant slope. After how man seconds does the soccer ball land on the hill? Check the reasonableness of the solution(s). You can use a sstem of equations to find the time the soccer ball lands on the hill. Step 1 Write an equation for each function. Use the model = 1 + v 0 + h 0 to represent the height of the soccer ball. The initial velocit v 0 is 30. From the graph, the initial height of the object h 0 is 0, so an equation is = The linear function has a -intercept of 0 and slope of 1, so an equation is =. Step Write a sstem of equations. = Equation 1 = Equation Step 3 Solve the sstem b substitution. Substitute for in Equation 1 and solve for = 0 ( 1 + 9) = 0 = Write Equation 1. = Substitute for. Write in standard form. Factor the polnomial. = 0 or = 0 Zero-Product Propert = 0 or 1.81 Solve for. Intersection X=1.815 Y= Reject the solution = 0 because this is when the ball is at the base of the hill. The ball lands on the hill about 1.81 seconds after it is kicked. Use a graphing calculator to check the reasonableness of the solution b graphing the sstem and using the intersect feature. Solving a Nonlinear Sstem b Elimination Solve the sstem b elimination. 5 = Equation = 0 Equation Check Add the equations to eliminate the -term and obtain a quadratic equation in. 5 = + + = = Add the equations = 0 Write in standard form. = 3 ± 15 Use the Quadratic Formula. Because the discriminant is negative, the equation = 0 has no real solution. So, the original sstem has no real solution. You can check this b graphing the sstem and seeing that the graphs do not appear to intersect. Section.5 Solving Nonlinear Sstems 185

45 Monitoring Progress Help in English and Spanish at BigIdeasMath.com Solve the sstem using an method. Eplain our choice of method. 1. = = = = = 5 + =. WHAT IF? In Eample, a second soccer ball is kicked with an initial velocit of 5 feet per second. After how man seconds does the soccer ball land on the hill? Center Radius: r Point on circle: (, ) Some nonlinear sstems have equations of the form + = r. This equation is the standard form of a circle with center (0, 0) and radius r. When the graphs of the equations in a sstem are a line and a circle, the graphs can intersect in zero, one, or two points. So, the sstem can have zero, one, or two solutions, as shown. No solution One solution Two solutions Solving a Nonlinear Sstem b Substitution Solve the sstem b substitution. + = 10 Equation 1 = Equation COMMON ERROR You can also substitute = 3 in Equation 1 to find. This ields two apparent solutions, (3, 1) and (3, 1). However, (3, 1) is not a solution because it does not satisf Equation. You can also see (3, 1) is not a solution from the graph. Substitute for in Equation 1 and solve for. + = 10 Write Equation 1. + ( ) = 10 Substitute for = = 0 Epand the power. Write in standard form. + 9 = 0 Divide each side b 10. ( 3) = 0 Perfect Square Trinomial Pattern = 3 To find the -coordinate of the solution, substitute = 3 in Equation. = 3(3) + 10 = 1 The solution is (3, 1). Check the solution b graphing the sstem. You can see that the line and the circle intersect onl at the point (3, 1). Zero-Product Propert Check (3, 1) 18 Chapter Quadratic Equations and Comple Numbers

46 Monitoring Progress Solve the sstem. Help in English and Spanish at BigIdeasMath.com 5. + = 1. + = 7. + = 1 = + = + = Solving Equations b Graphing You can solve an equation b rewriting it as a sstem of equations and then solving the sstem b graphing. Core Concept Solving Equations b Graphing Step 1 To solve the equation f () = g(), write a sstem of two equations, = f () and = g(). Step Graph the sstem of equations = f () and = g(). The -value of each solution of the sstem is a solution of the equation f () = g(). ANOTHER WAY In Eample 5, ou can also find the solutions b writing the given equation as + 3 = 0 and solving this equation using the Quadratic Formula. Solving Quadratic Equations b Graphing Solve = b graphing. Step 1 Write a sstem of equations using each side of the original equation. Equation Sstem = = = Step Use a graphing calculator to graph the sstem. Then use the intersect feature to find the -value of each solution of the sstem. 3 3 Intersection X= Y= Intersection X= Y= The graphs intersect when 1.18 and 0.3. The solutions of the equation are 1.18 and 0.3. Monitoring Progress Solve the equation b graphing. Help in English and Spanish at BigIdeasMath.com = ( 3) + 9. ( + )( 1) = Section.5 Solving Nonlinear Sstems 187

47 .5 Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check 1. WRITING Describe the possible solutions of a sstem consisting of two quadratic equations.. WHICH ONE DOESN T BELONG? Which sstem does not belong with the other three? Eplain our reasoning. = 3 + = + 1 = 1 = 3 + = = = = + 1 Monitoring Progress and Modeling with Mathematics In Eercises 3 10, solve the sstem b graphing. Check our solution(s). (See Eample 1.) 3. = +. = ( 3) + 5 = 0.5( + ) = 5 5. = = = 3 5 = = = 9 = 1 30 = 1 9. = ( ) 10. = 1 ( + ) = + = MODELING WITH MATHEMATICS The graph shows the parabolic surface of a skateboard ramp, where is the height (in inches) and is the horizontal distance (in inches). A spotlight is focused at a point on the ramp, represented b the linear function. How high is the spotlight focused on the ramp? Check the reasonableness of the solution(s). (See Eample.) 1. MODELING WITH MATHEMATICS The graph shows a linear function that represents the cost of producing a certain product and a quadratic function the represents the revenue of producing that product. How man products should be produced so that the cost is less than the revenue? Check the reasonableness of the solution(s). Value (dollars) Compan Profits (10, 5000) (10, 1300) (15, 3750) 1000 (0, 1000) (0, 0) Quantit In Eercises 13 1, solve the sstem b substitution. 0 (0, 5) = = = = (0, 8) (5,.8) (180, 38) 15. = = 7 = + + = 0 0 (100,.3) In Eercises 17, solve the sstem b elimination. (See Eample 3.) = = + = 5 + = = = Chapter Quadratic Equations and Comple Numbers

48 0. = 10 = = 1. = = 11 = = = = = 0 85 In Eercises 5 8, solve the sstem b substitution. (See Eample.) 5. + = 9. + = = 7 = = = = 1 + = 1 9. ERROR ANALYSIS Describe and correct the error in using elimination to solve a sstem. = = 1 0 = = 18 = NUMBER SENSE The table shows the inputs and outputs of two quadratic equations. Identif the solution(s) of the sstem. Eplain our reasoning In Eercises 31 3, solve the sstem using an method. Eplain our choice of method. 31. = 1 3. = 1 13 = = = 3( ) + ( ) + = = 100 = + 1 USING TOOLS In Eercises 37, solve the equation b graphing. (See Eample 5.) = = ( + )( ) = = ( ) 3 = ( + 3)( + 9) 38. ( + )( + 8) = ( + 3)( + 1) 1 3. REASONING A nonlinear sstem contains the equations of a constant function and a quadratic function. The sstem has one solution. Describe the relationship between the graphs.. PROBLEM SOLVING The range (in miles) of a broadcast signal from a radio tower is bounded b a circle given b the equation + = 10. A straight highwa can be modeled b the equation 1 = For what lengths of the highwa are cars able to receive the broadcast signal? 5. PROBLEM SOLVING A car passes a parked police car and continues at a constant speed r. The police car begins accelerating at a constant rate when it is passed. The diagram indicates the distance d (in miles) the police car travels as a function of time t (in minutes) after being passed. Write and solve a sstem of equations to find how long it takes the police car to catch up to the other car. t = 0 t =? r = 0.8 mi/min d =.5t = = = 10 = + 1 Section.5 Solving Nonlinear Sstems 189

49 . THOUGHT PROVOKING Write a nonlinear sstem that has two different solutions with the same -coordinate. Sketch a graph of our sstem. Then solve the sstem. 5. HOW DO YOU SEE IT? The graph of a nonlinear sstem is shown. Estimate the solution(s). Then describe the transformation of the graph of the linear function that results in a sstem with no solution. 7. OPEN-ENDED Find three values for m so the sstem has no solution, one solution, and two solutions. Justif our answer using a graph. 3 = = m MAKING AN ARGUMENT You and a friend solve the sstem shown and determine that = 3 and = 3. You use Equation 1 to obtain the solutions (3, 3), (3, 3), ( 3, 3), and ( 3, 3). Your friend uses Equation to obtain the solutions (3, 3) and ( 3, 3). Who is correct? Eplain our reasoning. 53. MODELING WITH MATHEMATICS To be eligible for a parking pass on a college campus, a student must live at least 1 mile from the campus center. + = 18 Equation 1 = 0 Equation 9. COMPARING METHODS Describe two different was ou could solve the quadratic equation. Which wa do ou prefer? Eplain our reasoning = ANALYZING RELATIONSHIPS Suppose the graph of a line that passes through the origin intersects the graph of a circle with its center at the origin. When ou know one of the points of intersection, eplain how ou can find the other point of intersection without performing an calculations. 51. WRITING Describe the possible solutions of a sstem that contains (a) one quadratic equation and one equation of a circle, and (b) two equations of circles. Sketch graphs to justif our answers. Maintaining Mathematical Proficienc 1 mi Main Street (0, 0) 1 mi mi campus center Oak Lane 5 mi College Drive a. Write equations that represent the circle and Oak Lane. b. Solve the sstem that consists of the equations in part (a). c. For what length of Oak Lane are students not eligible for a parking pass? 5. CRITICAL THINKING Solve the sstem of three equations shown. + = = + = + Reviewing what ou learned in previous grades and lessons Solve the inequalit. Graph the solution on a number line. (Skills Review Handbook) 55. > ( ) Write an inequalit that represents the graph. (Skills Review Handbook) Chapter Quadratic Equations and Comple Numbers

50 . TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..H Quadratic Inequalities Essential Question How can ou solve a quadratic inequalit? Work with a partner. The graphing calculator screen shows the graph of Solving a Quadratic Inequalit 3 f () = + 3. SELECTING TOOLS To be proficient in math, ou need to use technological tools to eplore our understanding of concepts. Eplain how ou can use the graph to solve the inequalit Then solve the inequalit. Solving Quadratic Inequalities Work with a partner. Match each inequalit with the graph of its related quadratic function. Then use the graph to solve the inequalit. 5 a. 3 + > 0 b c. 3 < 0 d. + 0 e. < 0 f. > 0 A. B. C. D. E. F. Communicate Your Answer 3. How can ou solve a quadratic inequalit?. Eplain how ou can use the graph in Eploration 1 to solve each inequalit. Then solve each inequalit. a. + 3 > 0 b. + 3 < 0 c Section. Quadratic Inequalities 191

52 Using a Quadratic Inequalit in Real Life A manila rope used for rappelling down a cliff can safel support a weight W (in pounds) provided W 180d where d is the diameter (in inches) of the rope. Graph the inequalit and interpret the solution. Graph W = 180d for nonnegative values of d. Because the inequalit smbol is, make the parabola solid. Test a point inside the parabola, such as (1, 3000). W 180d 3000? 180(1) Because (1, 3000) is not a solution, shade the region outside the parabola. The shaded region represents weights that can be supported b ropes with various diameters. Weight (pounds) Manila Rope W 3000 (1, 3000) W 180d d Diameter (inches) Graphing a sstem of quadratic inequalities is similar to graphing a sstem of linear inequalities. First graph each inequalit in the sstem. Then identif the region in the coordinate plane common to all of the graphs. This region is called the graph of the sstem. Graphing a Sstem of Quadratic Inequalities Graph the sstem of quadratic inequalities and identif two solutions of the sstem. Check Check that a point in the solution region, such as (0, 0), is a solution of the sstem. < <? < ? 0 + (0) < + 3 Inequalit Inequalit Step 1 Graph < + 3. The graph is the red region inside (but not including) the parabola = + 3. Step Graph + 3. The graph is the blue region inside and including the parabola = + 3. Step 3 Identif the purple region where the two graphs overlap. This region is the graph of the sstem. The points (0, 0) and ( 1, 1) are in the purple-shaded region. So, the are solutions of the sstem < Monitoring Progress Help in English and Spanish at BigIdeasMath.com. Graph the sstem of inequalities consisting of and > 3. State two solutions of the sstem. Section. Quadratic Inequalities 193

53 Solving Quadratic Inequalities in One Variable A quadratic inequalit in one variable can be written in one of the following forms, where a, b, and c are real numbers and a 0. a + b + c < 0 a + b + c > 0 a + b + c 0 a + b + c 0 You can solve quadratic inequalities using algebraic methods or graphs. Solve 3 < 0 algebraicall. Solving a Quadratic Inequalit Algebraicall First, write and solve the equation obtained b replacing < with =. 3 = 0 Write the related equation. ( )( + 1) = 0 Factor. = or = 1 Zero-Product Propert The numbers 1 and are the critical values of the original inequalit. Plot 1 and on a number line, using open dots because the values do not satisf the inequalit. The critical -values partition the number line into three intervals. Test an -value in each interval to determine whether it satisfies the inequalit Test =. Test = 0. Test = 5. ( ) 3( ) = < 0 0 3(0) = < 0 5 3(5) = < 0 So, the solution is 1 < <. Another wa to solve a + b + c < 0 is to first graph the related function = a + b + c. Then, because the inequalit smbol is <, identif the -values for which the graph lies below the -ais. You can use a similar procedure to solve quadratic inequalities that involve, >, or. Solve b graphing. Solving a Quadratic Inequalit b Graphing = 3 5 The solution consists of the -values for which the graph of = 3 5 lies on or above the -ais. Find the -intercepts of the graph b letting = 0 and using the Quadratic Formula to solve 0 = 3 5 for. = ( 1) ± ( 1) (3)( 5) (3) a = 3, b = 1, c = 5 = 1 ± 1 Simplif. The solutions are 1.1 and 1.7. Sketch a parabola that opens up and has 1.1 and 1.7 as -intercepts. The graph lies on or above the -ais to the left of (and including) = 1.1 and to the right of (and including) = 1.7. The solution of the inequalit is approimatel 1.1 or Chapter Quadratic Equations and Comple Numbers

54 Modeling with Mathematics A rectangular parking lot must have a perimeter of 0 feet and an area of at least 8000 square feet. Describe the possible lengths of the parking lot. ANOTHER WAY You can graph each side of 0 = 8000 and use the intersection points to determine when 0 is greater than or equal to Understand the Problem You are given the perimeter and the minimum area of a parking lot. You are asked to determine the possible lengths of the parking lot.. Make a Plan Use the perimeter and area formulas to write a quadratic inequalit describing the possible lengths of the parking lot. Then solve the inequalit. 3. Solve the Problem Let represent the length (in feet) and let w represent the width (in feet) of the parking lot. Perimeter = 0 Area w = 0 w 8000 Solve the perimeter equation for w to obtain w = 0. Substitute this into the area inequalit to obtain a quadratic inequalit in one variable. w 8000 Write the area inequalit. (0 ) 8000 Substitute 0 for w Distributive Propert Write in standard form. Use a graphing calculator to find the -intercepts of = USING TECHNOLOGY Variables displaed when using technolog ma not match the variables used in applications. In the graphs shown, the length corresponds to the independent variable and the area A corresponds to the dependent variable. Zero X= Y= Zero X= Y= The -intercepts are 5.97 and The solution consists of the -values for which the graph lies on or above the -ais. The graph lies on or above the -ais when So, the approimate length of the parking lot is at least feet and at most 17 feet.. Look Back Choose a length in the solution region, such as = 100, and find the width. Then check that the dimensions satisf the original area inequalit. + w = 0 w 8000 (100) + w = 0 100(110)? 8000 w = , Monitoring Progress Solve the inequalit. Help in English and Spanish at BigIdeasMath.com < > 5 8. WHAT IF? In Eample, the area must be at least 8500 square feet. Describe the possible lengths of the parking lot. Section. Quadratic Inequalities 195

55 . Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check 1. WRITING Compare the graph of a quadratic inequalit in one variable to the graph of a quadratic inequalit in two variables.. WRITING Eplain how to solve + 8 < 0 using algebraic methods and using graphs. Monitoring Progress and Modeling with Mathematics In Eercises 3, match the inequalit with its graph. Eplain our reasoning > < A. B. ERROR ANALYSIS In Eercises 17 and 18, describe and correct the error in graphing C. D In Eercises 7 1, graph the inequalit. (See Eample 1.) 7. < > < > ( + 3) 1 1. ( 1 ) + 5 ANALYZING RELATIONSHIPS In Eercises 15 and 1, use the graph to write an inequalit in terms of f () so point P is a solution. 15. P = f() 1. = f() P 19. MODELING WITH MATHEMATICS A hardwood shelf in a wooden bookcase can safel support a weight W (in pounds) provided W 115, where is the thickness (in inches) of the shelf. Graph the inequalit and interpret the solution. (See Eample.) 0. MODELING WITH MATHEMATICS A wire rope can safel support a weight W (in pounds) provided W 8000d, where d is the diameter (in inches) of the rope. Graph the inequalit and interpret the solution. In Eercises 1, graph the sstem of quadratic inequalities and state two solutions of the sstem. (See Eample 3.) 1.. > 5 < + 1 > < < Chapter Quadratic Equations and Comple Numbers

56 In Eercises 7 3, solve the inequalit algebraicall. (See Eample.) 7. < < > > MODELING WITH MATHEMATICS The arch of the Sdne Harbor Bridge in Sdne, Australia, can be modeled b = , where is the distance (in meters) from the left plons and is the height (in meters) of the arch above the water. For what distances is the arch above the road? plon In Eercises 35, solve the inequalit b graphing. (See Eample 5.) < > 0 5 m > < < DRAWING CONCLUSIONS Consider the graph of the function f() = a + b + c.. PROBLEM SOLVING The number T of teams that have participated in a robot-building competition for high-school students over a recent period of time (in ears) can be modeled b T() = , 0. After how man ears is the number of teams greater than 1000? Justif our answer. 1 a. What are the solutions of a + b + c < 0? b. What are the solutions of a + b + c > 0? c. The graph of g represents a reflection in the -ais of the graph of f. For which values of is g() positive?. MODELING WITH MATHEMATICS A rectangular fountain displa has a perimeter of 00 feet and an area of at least 9100 feet. Describe the possible widths of the fountain. (See Eample.) 7. PROBLEM SOLVING A stud found that a driver s reaction time A() to audio stimuli and his or her reaction time V() to visual stimuli (both in milliseconds) can be modeled b A() = , 1 70 V() = , 1 70 where is the age (in ears) of the driver. a. Write an inequalit that ou can use to find the -values for which A() is less than V(). b. Use a graphing calculator to solve the inequalit A() < V(). Describe how ou used the domain 1 70 to determine a reasonable solution. c. Based on our results from parts (a) and (b), do ou think a driver would react more quickl to a traffic light changing from green to ellow or to the siren of an approaching ambulance? Eplain. Section. Quadratic Inequalities 197

57 8. HOW DO YOU SEE IT? The graph shows a sstem of quadratic inequalities MATHEMATICAL CONNECTIONS The area A of the region bounded b a parabola and a horizontal line can be modeled b A = bh, where b and h are as 3 defined in the diagram. Find the area of the region determined b each pair of inequalities. 8 8 b h a. Identif two solutions of the sstem. b. Are the points (1, ) and (5, ) solutions of the sstem? Eplain. c. Is it possible to change the inequalit smbol(s) so that one, but not both of the points, is a solution of the sstem? Eplain. 9. MODELING WITH MATHEMATICS The length L (in millimeters) of the larvae of the black porg fish can be modeled b L() = , 0 0 where is the age (in das) of the larvae. Write and solve an inequalit to find at what ages a larva s length tends to be greater than 10 millimeters. Eplain how the given domain affects the solution. a. + b THOUGHT PROVOKING Draw a compan logo that is created b the intersection of two quadratic inequalities. Justif our answer. 53. REASONING A truck that is 11 feet tall and 7 feet wide is traveling under an arch. The arch can be modeled b = , where and are measured in feet. 50. MAKING AN ARGUMENT You claim the sstem of inequalities below, where a and b are real numbers, has no solution. Your friend claims the sstem will alwas have at least one solution. Who is correct? Eplain. < ( + a) < ( + b) Maintaining Mathematical Proficienc Graph the function. Label the -intercept(s) and the -intercept. (Section 3.) a. Will the truck fit under the arch? Eplain. b. What is the maimum width that a truck 11 feet tall can have and still make it under the arch? c. What is the maimum height that a truck 7 feet wide can have and still make it under the arch? 5. f () = ( + 7)( 9) 55. g() = ( ) 5. h() = + 5 Find the minimum value or maimum value of the function. Then describe where the function is increasing and decreasing. (Section 3.) 57. f () = h() = 1 ( + ) f () = ( 3)( + 7) 0. h() = Reviewing what ou learned in previous grades and lessons 198 Chapter Quadratic Equations and Comple Numbers

58 .. What Did You Learn? Core Vocabular Quadratic Formula, p. 17 discriminant, p. 17 sstem of nonlinear equations, p. 18 quadratic inequalit in two variables, p. 19 quadratic inequalit in one variable, p. 19 Core Concepts Section. Solving Equations Using the Quadratic Formula, p. 17 Analzing the Discriminant of a + b + c = 0, p. 17 Methods for Solving Quadratic Equations, p. 177 Modeling Launched Objects, p. 178 Section.5 Solving Sstems of Nonlinear Equations, p. 18 Solving Equations b Graphing, p. 187 Section. Graphing a Quadratic Inequalit in Two Variables, p. 19 Solving Quadratic Inequalities in One Variable, p. 19 Mathematical Thinking 1. How can ou use technolog to determine whose rocket lands first in part (b) of Eercise 5 on page 181?. What question can ou ask to help the person avoid making the error in Eercise 8 on page 190? 3. Eplain our plan to find the possible widths of the fountain in Eercise on page 197. Some people have attached earlobes, the recessive trait. Some people have free earlobes, the dominant trait. What percent of people carr both traits? To eplore the answer to this question and more, go to BigIdeasMath.com. Performance Task Algebra in Genetics: The Hard-Weinberg Law 199

59 Chapter Review.1 Solving Quadratic Equations (pp ) In a phsics class, students must build a Rube Goldberg machine that drops a ball from a 3-foot table. Write a function h (in feet) of the ball after t seconds. How long is the ball in the air? The initial height is 3, so the model is h = 1t + 3. Find the zeros of the function. h = 1t + 3 Write the function. 0 = 1t + 3 Substitute 0 for h. 3 = 1t Subtract 3 from each side. 3 1 = t Divide each side b 1. 3 ± = t Take square root of each side. 1 ±0.3 t Use a calculator. Reject the negative solution, 0.3, because time must be positive. The ball will fall for about 0.3 second before it hits the ground. 1. Solve 8 = 0 b graphing. Solve the equation using square roots or b factoring.. 3 = = = A rectangular enclosure at the zoo is 35 feet long b 18 feet wide. The zoo wants to double the area of the enclosure b adding the same distance to the length and width. Write and solve an equation to find the value of. What are the dimensions of the enclosure?. Comple Numbers (pp ) Perform each operation. Write the answer in standard form. a. (3 i ) (7 + i ) = (3 7) + ( )i b. 5i( + 5i ) = 0i + 5i = 8i = 0i + 5( 1) = 5 + 0i. Find the values and that satisf the equation 3 i = + 3i. Perform the operation. Write the answer in standard form. 7. ( + 3i ) + (7 i ) 8. (9 + 3i ) ( 7i ) 9. (5 + i )( + 7i ) 10. Solve = Find the zeros of f () = Chapter Quadratic Equations and Comple Numbers

60 .3 Completing the Square (pp ) Solve = 0 b completing the square = 0 Write the equation. + 1 = = Add ( b Write left side in the form + b. ) = ( 1 ) = 3 to each side. ( + ) = 8 Write left side as a binomial squared. + = ± 8 = ± 8 = ± 7 The solutions are = + 7 and = 7. Take square root of each side. Subtract from each side. Simplif radical. 1. An emploee at a local stadium is launching T-shirts from a T-shirt cannon into the crowd during an intermission of a football game. The height h (in feet) of the T-shirt after t seconds can be modeled b h = 1t + 9t +. Find the maimum height of the T-shirt. Solve the equation b completing the square = = ( ) = Write = + 0 in verte form. Then identif the verte.. Using the Quadratic Formula (pp ) Solve + = 5 using the Quadratic Formula. + = 5 Write the equation. + 5 = 0 = ± ( 1)( 5) ( 1) = ± ± i = = ± i The solutions are + i and i. Solve the equation using the Quadratic Formula. Write in standard form. a = 1, b =, c = 5 Simplif. Write in terms of i. Simplif = = = 0 Find the discriminant of the quadratic equation and describe the number and tpe of solutions of the equation = = = 0 Chapter Chapter Review 01

61 .5 Solving Nonlinear Sstems (pp ) Solve the sstem b elimination. 8 + = 5 Equation 1 1 = 31 Equation Add the equations to eliminate the -term and obtain a quadratic equation in. 8 + = 5 1 = 31 = 3 Add the equations. + 3 = 0 Write in standard form. + 9 = 0 Divide each side b. ( 3) = 0 Perfect Square Trinomial Pattern = 3 Zero-Product Propert To solve for, substitute = 3 in Equation 1 to obtain = 1. So, the solution is (3, 1). Solve the sstem b an method. Eplain our choice of method. 3. = = 5. + = + = = = 5. Solve = b graphing.. Quadratic Inequalities (pp ) Graph the sstem of quadratic inequalities. > Inequalit Inequalit Step 1 Graph >. The graph is the red region inside (but not including) the parabola =. Step Graph 3 +. The graph is the blue region inside and including the parabola = 3 +. Step 3 Identif the purple region where the two graphs overlap. This region is the graph of the sstem. Graph the inequalit. 7. > Graph the sstem of quadratic inequalities > > > Solve the inequalit < Chapter Quadratic Equations and Comple Numbers

62 Chapter Test Solve the equation using an method. Provide a reason for our choice = = = 85. ( + )( 1) = Eplain how to use the graph to find the number and tpe of solutions of the quadratic equation. Justif our answer b using the discriminant = = = Solve the sstem of equations or inequalities = = + 0 = 18 < ( + 3) + = Write (3 + i )( i ) as a comple number in standard form. 1. The aspect ratio of a widescreen TV is the ratio of the screen s width to its height, or 1 : 9. What are the width and the height of a 3-inch widescreen TV? Justif our answer. (Hint: Use the Pthagorean Theorem and the fact that TV sizes refer to the diagonal length of the screen.) 3 in The shape of the Gatewa Arch in St. Louis, Missouri, can be modeled b = , where is the distance (in feet) from the left foot of the arch and is the height (in feet) of the arch above the ground. For what distances is the arch more than 00 feet above the ground? Justif our answer. 1. You are plaing a game of horseshoes. One of our tosses is modeled in the diagram, where is the horseshoe s horizontal position (in feet) and is the corresponding height (in feet). Find the maimum height of the horseshoe. Then find the distance the horseshoe travels. Justif our answer. = Chapter Chapter Test 03

63 Standards Assessment 1. Which quadratic equation has the roots = 1 + 3i and = 1 3i? (TEKS A.7.A) A + 10 = 0 B = 0 C + = 8 D = 8. Which quadratic function is represented b the graph? (TEKS A..B) F = G = H = ( 3) + J none of the above 3. GRIDDED ANSWER The length of the hpotenuse of a right triangle is 9 inches. The length of the longer leg is 1 inch greater than the length of the shorter leg. What is the length (in inches) of the longer leg? (TEKS A.3.A, TEKS A.3.C). Each ear, an engineering firm emplos senior interns and junior interns. Senior interns receive \$50 per week and junior interns receive \$350 per week. This ear, a minimum of 7 but no more than 13 interns will be hired. Due to budgetar constraints, the amount spent on interns cannot eceed \$5000. Which statement is not true? (TEKS A.3.G) A The firm can hire 13 junior interns. B The firm can hire senior interns and 7 junior interns. C When both tpes of interns are hired, the firm will spend a minimum of \$550 per week on interns. D When both tpes of interns are hired, the firm will spend a maimum of \$5850 per week on interns. 5. A deliver service guideline states that for a rectangular package such as the one shown, the sum of the length and the girth cannot eceed 108 inches. The length of a rectangular package is 3 inches. What is the maimum possible volume? (TEKS A..F) w girth h length F 11, in. G 11, in. 3 H, in. 3 J,5 in. 3 0 Chapter Quadratic Equations and Comple Numbers

64 . Use the data in the table to find the most reasonable estimate of when = 13. (TEKS A.8.A, TEKS A.8.C) A B 8 C 85 D What is the value of k when the function = is written in the form = a( h) + k? (TEKS A..D) F 18 G 30 H 3 J What is an equation of the parabola with focus at ( 8, 0) and verte at (0, 0)? (TEKS A..B) A = 0.5 B = 3 C = 8 D = 3 9. Solve the quadratic inequalit (TEKS A..H) F 0 or.5 G 0.5 H 8 or 3.5 J The figure shows the temperatures (in degrees Celsius) at various locations on a metal plate. The temperature at an interior junction, such as T 1, T, or T 3, can be approimated b finding the mean of the temperatures at the four surrounding junctions. Which of the following statements are true? (TEKS A.3.A, TEKS A.3.B) I. T 1 T T 3 = 80 II. T = 37.5 C III. T 3 = 5 C 0 T T 50 0 T A I and II onl B I and III onl C II and III onl D I, II, and III 11. Find the solutions of the equation = 5. (TEKS A..F) F = 5 ± 3 G = 5 ± 3 H = 5 ± 3 J = 5 ± 3 Chapter Standards Assessment 05

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