4 B. 4 D. 4 F. 3. How can you use the graph of a quadratic equation to determine the number of real solutions of the equation?

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1 3.1 Solving Quadratic Equations COMMON CORE Learning Standards HSA-SSE.A. HSA-REI.B.b HSF-IF.C.8a Essential Question Essential Question How can ou use the graph of a quadratic equation to determine the number of real solutions of the equation? Matching a Quadratic Function with Its Graph Work with a partner. Match each quadratic function with its graph. Eplain our reasoning. Determine the number of -intercepts of the graph. a. f () = b. f () = + 1 c. f () = + d. f () = + e. f () = + 1 f. f () = + A. B. C. D. E. F. Solving Quadratic Equations MAKING SENSE OF PROBLEMS To be proficient in math, ou need to make conjectures about the form and meaning of solutions. Work with a partner. Use the results of Eploration 1 to find the real solutions (if an) of each quadratic equation. a. = 0 b. + 1 = 0 c. + = 0 d. + = 0 e. + 1 = 0 f. + = 0 Communicate Your Answer 3. How can ou use the graph of a quadratic equation to determine the number of real solutions of the equation?. How man real solutions does the quadratic equation = 0 have? How do ou know? What are the solutions? Section 3.1 Solving Quadratic Equations 93

2 3.1 Lesson Core Vocabular quadratic equation in one variable, p. 9 root of an equation, p. 9 zero of a function, p. 9 Previous properties of square roots factoring rationalizing the denominator STUDY TIP Quadratic equations can have zero, one, or two real solutions. What You Will Learn Solve quadratic equations b graphing. Solve quadratic equations algebraicall. Solve real-life problems. Solving Quadratic Equations b Graphing A quadratic equation in one variable is an equation that can be written in the standard form a + b + c = 0, where a, b, and c are real numbers and a 0. A root of an equation is a solution of the equation. You can use various methods to solve quadratic equations. Core Concept Solving Quadratic Equations B graphing Find the -intercepts of the related function = a + b + c. Using square roots Write the equation in the form u = d, where u is an algebraic epression, and solve b taking the square root of each side. B factoring Write the polnomial equation a + b + c = 0 in factored form and solve using the Zero-Product Propert. Solving Quadratic Equations b Graphing Solve each equation b graphing. a. = 0 b. = Check = 0 ( ) ( ) =? 0 + =? 0 0 = 0 = =? =? 0 0 = 0 a. The equation is in standard form. b. Add to each side to obtain Graph the related function = 0. Graph the =. related function =. ( 1, 0) 8 (3, 0) 8 1 (, 0) The -intercepts are and 3. The -intercept is 1. The solutions, or roots, The solution, or root, is are = and = 3. = 1. Monitoring Progress Solve the equation b graphing. Help in English and Spanish at BigIdeasMath.com = = = 0 9 Chapter 3 Quadratic Equations and Comple Numbers

3 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 LOOKING FOR STRUCTURE 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 3.1 Solving Quadratic Equations 95

4 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 3 Quadratic Equations and Comple Numbers

5 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 3.1 Solving Quadratic Equations 97

6 Height (feet) Height of Dropped Object h 1 3 t Time (seconds) INTERPRETING EXPRESSIONS 0 sec, 00 ft 1 sec, 18 ft sec, 13 ft 3 sec, 5 ft 3.5 sec, 0 ft 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 98 Chapter 3 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)?

7 3.1 Eercises Dnamic Solutions available 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 3.1 Solving Quadratic Equations 99

8 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). 100 Chapter 3 Quadratic Equations and Comple Numbers

9 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 3.1 Solving Quadratic Equations 101

10 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) 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. 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 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 epanded part of lot old lot 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) 15 ft 7. ( + ) + ( ) 77. ( 3 + ) + (3 + 10) 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 10 Chapter 3 Quadratic Equations and Comple Numbers

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