CHAPTER The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities 009 Carnegie Learning, Inc. The Chinese invented rockets over 700 years ago. Since then rockets have been used to propel fireworks, weaponry, cargo, and even humans off the Earth s surface. You will learn to use a quadratic function to model the height of a launched object over time..1 Quadratic Formula Deriving the Quadratic Formula p. 99. Calculating Roots and Zeros Solving Quadratic Equations and Inequalities p. 111.3 The Discriminant The Discriminant and the Nature of Roots/Vertex Form p. 11.4 The Complete Number System Operations with Complex Numbers p. 131.5 Complex Roots and Zeros Complex Roots and Simplifying Complex Roots p. 137.6 Cases, Roots, and Graphs Solving Quadratic Inequalities p. 145.7 Carl Freidrich Gauss: Child Prodigy Arithmetic Sequences, Series, and Their Partial Sums p. 153.8 Mathematics Empowers Us To Do Superhuman Mental Calculations! Modeling Partial Sums of Arithmetic Series p. 163 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities 97
009 Carnegie Learning, Inc. 98 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
.1 Quadratic Formula Deriving the Quadratic Formula Objectives Key Term In this lesson, you will: Quadratic Formula Derive the quadratic formula. Use the quadratic formula to solve quadratic equations and inequalities. Use the quadratic formula to find zeros of quadratic functions. Problem 1 For each quadratic function in standard form: a. Calculate the y-intercept. b. Calculate the equation for the axis of symmetry. c. Calculate the coordinates of the vertex. d. Calculate the x-intercept(s). e. Calculate the coordinates of the point on the parabola symmetric to the y-intercept. 009 Carnegie Learning, Inc. f. Sketch a graph of the function with parts (a) through (e) indicated on the graph. 1. f(x) x 7x 1 a. y-intercept: b. Axis of symmetry: c. Vertex: Lesson.1 Deriving the Quadratic Formula 99
d. x-intercept(s): e. Point symmetric to the y-intercept: f. Graph:. f(x) x 11x 1 a. y-intercept: b. Axis of symmetry: 009 Carnegie Learning, Inc. c. Vertex: 100 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
d. x-intercept(s): e. Point symmetric to the y-intercept: 009 Carnegie Learning, Inc. f. Graph: Lesson.1 Deriving the Quadratic Formula 101
3. f(x) 3x 11x a. y-intercept: b. Axis of symmetry: c. Vertex: d. x-intercept(s): e. Point symmetric to the y-intercept: f. Graph: 009 Carnegie Learning, Inc. 4. f(x) 5x 10x 6 a. y-intercept: b. Axis of symmetry: 10 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
c. Vertex: d. x-intercept(s): e. Point symmetric to the y-intercept: f. Graph: 009 Carnegie Learning, Inc. 5. Does the graph of f(x) 3x 11x in Question 3 have x-intercepts? Were you able to calculate them? Why or why not? 6. Does the graph of f(x) 5x 10x 6 in Question 4 have x-intercepts? Were you able to calculate them? Why or why not? Lesson.1 Deriving the Quadratic Formula 103
Problem For each quadratic function in standard form: a. Calculate the coordinates of the vertex. b. Calculate the equation for the axis of symmetry. c. Calculate the y-intercept. d. Calculate the x-intercept(s). e. Calculate the coordinates of the point on the parabola symmetric to the y-intercept. f. Sketch a graph of the function with parts (a) through (e) indicated on the graph. 1. f(x) (x 5) 4 a. Vertex: b. Axis of symmetry: c. y-intercept: d. x-intercept(s): e. Point symmetric to the y-intercept: 009 Carnegie Learning, Inc. 104 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
f. Graph:. f(x) (x 3) 1 a. Vertex: b. Axis of symmetry: c. y-intercept: d. x-intercept(s): 009 Carnegie Learning, Inc. e. Point symmetric to the y-intercept: Lesson.1 Deriving the Quadratic Formula 105
f. Graph: 3. Does the graph of f(x) (x 3) 1 in Question have x-intercepts? Were you able to calculate them? Why or why not? Problem 3 Calculating the x-intercepts for many quadratic functions can be difficult if the quadratic function cannot be factored. In this problem we will derive a method for calculating the zeros of quadratic functions that do not factor. Recall that the vertex of a quadratic function in standard form y ax bx c is: ( ( or, ba b 4ac, f ba ( ba )) 4a ) ( 1. Write a quadratic function in vertex form that has a vertex at,. ba b 4ac 4a ) A formula for the x-intercepts of any quadratic function can be derived by setting this function equal to zero and solving for x. f(x) a ( x b a ) b 4ac 4a a ( x b a ) b 4ac 4a 0 009 Carnegie Learning, Inc. 106 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
a ( x b a ) b 4ac 4a a ( x b a ) a b 4ac 4a a ( x b a ) b 4ac 4a ( x b a ) b 4ac 4a x b a b 4ac a x b a b a b 4ac a b a x b a b 4ac a x b b 4ac a This formula is called the Quadratic Formula. It can be used to calculate the roots of any quadratic equation and the zeros of any quadratic function.. Use the Quadratic Formula to verify the zeros that you calculated in Problems 1 and. Then use the Quadratic Formula to determine the zeros you could not calculate. a. From Problem 1 Question 1: f(x) x 7x 1 009 Carnegie Learning, Inc. Lesson.1 Deriving the Quadratic Formula 107
b. From Problem 1 Question : f(x) x 11x 1 c. From Problem Question 1: f(x) (x 5) 4 d. From Problem 1 Question 3: f(x) 3x 11x 009 Carnegie Learning, Inc. 108 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
e. From Problem Question : f(x) (x 3) 1 Be prepared to share your methods and solutions. 009 Carnegie Learning, Inc. Lesson.1 Deriving the Quadratic Formula 109
009 Carnegie Learning, Inc. 110 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
. Calculating Roots and Zeros Solving Quadratic Equations and Inequalities Objectives In this lesson, you will: Use the Quadratic Formula to solve quadratic equations. Use the Quadratic Formula to calculate the zeros of quadratic functions. Use the Quadratic Formula to solve quadratic inequalities. Problem 1 Use the Quadratic Formula to solve each quadratic equation. 009 Carnegie Learning, Inc. 1. x 11x 10 0 Lesson. Solving Quadratic Equations and Inequalities 111
. x x 7 0 3. x 7x 3 0 009 Carnegie Learning, Inc. 11 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
4. 5x 11 0 Problem Use the Quadratic Formula to calculate the zeros of each quadratic function. 1. f(x) 4x 7x 8 009 Carnegie Learning, Inc. Lesson. Solving Quadratic Equations and Inequalities 113
. y 3x 11x 4 3. g(x) x 5x 7 009 Carnegie Learning, Inc. 114 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
4. y.4x 3x 5. Problem 3 Previously, we used quadratic functions to model vertical motion. 009 Carnegie Learning, Inc. A projectile is shot straight up into the air with an initial velocity of 500 ft/sec from 5 feet off the ground. Its height can be modeled with the quadratic function h(t) 16t 500t 5, where t is the time in seconds and h(t) is the height of the projectile in feet. 1. Which part of the parabola represents the maximum height of the projectile?. How long after the projectile is shot does it reach its maximum height? 3. What is the maximum height of the projectile? Lesson. Solving Quadratic Equations and Inequalities 115
4. How long does it take for the projectile to return to the ground? 5. Calculate when the projectile is 000 feet from the ground. 009 Carnegie Learning, Inc. 116 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
6. Write a quadratic inequality that represents the times when the projectile is higher than 000 feet. 7. When is the projectile higher than 000 feet? 8. When is the projectile below 000 feet? Problem 4 The solution to a quadratic inequality is the set of values that satisfy the inequality. To solve a quadratic inequality, perform the following: Calculate the roots of the corresponding quadratic equation. Test values that are greater than, between, and less than the roots to determine which intervals satisfy the inequality. Solve the inequality x 4x 5. Solve the corresponding quadratic equation: x 4x 5 x 4x 3 0 009 Carnegie Learning, Inc. (x 3)(x 1) 0 x 3 or x 1 Check: (3) 4(3) 3 9 1 3 0 (1) 4(1) 3 1 4 3 0 Pick a value less than 1, a value between 1 and 3, and a value greater than 3 to determine which intervals satisfy the inequality. Lesson. Solving Quadratic Equations and Inequalities 117
Try x 0, x, and x 4. (0) 4(0) not less than 5 () 4() 6 is less than 5 (4) 4(4) not less than 5 The only interval that satisfies the original inequality is the interval between 1 and 3. The inequality is less than or equal to, so 1 and 3 are included in the solution. Solution: x [1, 3] Solve each quadratic inequality. 1. 3x 5x 8 4 009 Carnegie Learning, Inc. 118 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
. 5x 10x 7 13 Be prepared to share your methods and solutions. 009 Carnegie Learning, Inc. Lesson. Solving Quadratic Equations and Inequalities 119
009 Carnegie Learning, Inc. 10 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
.3 The Discriminant The Discriminant and the Nature of Roots/Vertex Form Objectives Key Terms In this lesson, you will: Classify the number and nature of the roots/zeros of quadratic equations/ functions. Use the discriminant to classify the roots/zeros of quadratic equations/functions. Solve for the vertex form of quadratic functions. discriminant vertex form of a quadratic function Problem 1 Use the Quadratic Formula to calculate the zero(s) of each quadratic function. Next, solve for the average of the zeros to identify each function s vertex and axis of symmetry. Then, graph all three functions in the grid shown. 1. y x 009 Carnegie Learning, Inc. a. Zero(s): b. Axis of symmetry: c. Vertex: Lesson.3 The Discriminant and the Nature of Roots/Vertex Form 11
. y x a. Zero(s): b. Axis of symmetry: c. Vertex: 3. y x 4 a. Zero(s): b. Axis of symmetry: c. Vertex: 009 Carnegie Learning, Inc. 4. What are the similarities among the graphs? The axes of symmetry? The vertices? 1 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
5. How many zeros does each of the three functions have? a. Examine the Quadratic Formula. What portion of the formula determines the number of zeros? How do you know? b. Because this portion of the formula discriminates the number and nature of the roots, it is called the discriminant. Using the discriminant, write three inequalities to describe when a quadratic function has i. no real roots/zeros. ii. one real root/zero. iii. two real roots/zeros. 6. Use the discriminant to determine the number of roots/zeros that each equation/ function has. Then solve for the roots/zeros. a. y x 1x 009 Carnegie Learning, Inc. b. 0 x 1x 0 c. y x 1x 36 Lesson.3 The Discriminant and the Nature of Roots/Vertex Form 13
d. y 3x 7x 0 e. y 4x 9 f. 0 9x 1x 4 7. Examine each of the roots/zeros of the equations/functions in Question 6. Describe the nature of the roots. In other words, what characteristic of the discriminant determines if the roots will be rational or irrational? 8. Based on the number and nature of each of the following roots/zeros, decide if (i) the discriminant is positive, negative, or zero and (ii) the discriminant is or is not a perfect square. a. no real roots/zeros i. ii. 009 Carnegie Learning, Inc. b. one rational root/zero i. ii. 14 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
c. two rational roots/zeros i. ii. d. two irrational roots/zeros i. ii. Problem From previous activities, we know that all quadratic functions with the same leading coefficient, a, have the same shape. When graphing any quadratic function with a leading coefficient of a, we can solve for the vertex using the formula x b and a then graph the function, y ax, using the vertex as the initial point. 1. For each of the following quadratic functions, sketch a graph using the given vertex. a. y 4x Vertex: ( 5 ) 0x 5, 0 009 Carnegie Learning, Inc. Lesson.3 The Discriminant and the Nature of Roots/Vertex Form 15
b. y x 8x 1 Vertex: (4, 4) c. y 4x 4 Vertex: (0, 4). Factor each of the corresponding quadratic equations. a. 4x 0x 5 0 b. x 8x 1 0 c. 4x 4 0 009 Carnegie Learning, Inc. 16 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
3. For each of the following, complete the square without changing the function. Then solve for the vertex. An example is shown: 9x 18x 5 y 9(x x) 5 y 9(x x 1) 5 9 y b a ( 18) 18 Vertex: (1, 16) 1 9(1) 18(1) 5 9 18 5 16 y 9(x 1) 16 y a. 4x 0x 5 y b. x 8x 1 y c. 4x 4 y 009 Carnegie Learning, Inc. d. x 6x 8 y Lesson.3 The Discriminant and the Nature of Roots/Vertex Form 17
e. 3x 1 x 8 y 4. What do you notice about the final form of the quadratic function and the coordinates of the vertex? 5. The form y a(x h) k is called the vertex form of a quadratic function. In terms of this form, what are the coordinates of the vertex? a. What does a tell you about the graph of this function? b. What do h and k tell you about the graph of the quadratic function? 6. For each of the following quadratic functions in vertex form, determine its vertex and sketch its graph. a. y (x 3) 4 009 Carnegie Learning, Inc. 18 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
b. y (x ) 3 c. y 3(x 1) 5 009 Carnegie Learning, Inc. 7. Rewrite each of the following quadratic functions in vertex form. a. y 3x 6x 4 Lesson.3 The Discriminant and the Nature of Roots/Vertex Form 19
b. y 5x 15x c. y 4x 16x 1 8. Describe how the variables of the vertex form of a quadratic function tell you how the graph of the function will be transformed. Be prepared to share your work with another pair, group, or the entire class. 009 Carnegie Learning, Inc. 130 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
.4 The Complete Number System Operations with Complex Numbers Objectives In this lesson, you will: Add, subtract, multiply, and divide complex numbers. Determine the conjugate of a complex number. Calculate powers and roots of complex numbers. Key Terms conjugate of a complex number power of a complex number root of a complex number Problem 1 Adding, Subtracting, and Multiplying Complex Numbers To add and subtract complex numbers, combine the real terms and then combine the imaginary terms to form an answer that consists of two terms. 1. For each of the following pairs of complex numbers, calculate the sum and the difference. a. 5 3i, 6 5i 009 Carnegie Learning, Inc. b. 1 4i, 11 7i c. 8i, 11 17i d. 1 1.3i,.6 7.6i Multiplying complex numbers is similar to multiplying binomials. Use the distributive property twice so that each term of the first complex number is multiplied by each term of the second complex number. Lesson.4 Operations with Complex Numbers 131
. For each of the pairs of complex numbers in Question 1, calculate the product. a. b. c. d. Problem Dividing Complex Numbers Division of complex numbers requires using the conjugate of a complex number, thus changing the divisor into a real number. The conjugate of a complex number a bi is a bi. 1. For each of the following complex numbers, write its conjugate. a. 7 i b. 5 3i c. 1 11i d. 4i f. a bi. For each complex number and its conjugate in Question 1, calculate the product. a. b. c. d. e. f. 13 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities 009 Carnegie Learning, Inc. e. 9 7i
3. In each case, what happened when you multiplied the complex number (a bi ) by its conjugate (a bi)? Explain. Using this fact, we can simplify the division of a complex number by multiplying both the divisor and the dividend by the conjugate of the divisor, thus changing the divisor into a real number. For example: 3 i 3 i 4 3i 4 3i 4 3i 4 3i 1 9i 8i 6i 16 1i 1i 9i 1 17i 6 16 9 6 17i 5 3 i 4 3i 6 17 5 5 i 4. Calculate the following quotients. a. b. c. i 3 i 3 4i 3i 5 i 1 i 009 Carnegie Learning, Inc. d. 0 5i 4i Lesson.4 Operations with Complex Numbers 133
Problem 3 Calculating Powers and Roots of Complex Numbers Calculating a whole number power of a complex number can be accomplished by repeated multiplication, but this process is very time consuming. 1. Calculate the indicated power of each of the following complex numbers. a. ( 3i ) b. ( 1 i ) 3 c. ( 1 3i ) 3 Calculating the square root of a complex number can be accomplished using straightforward algebraic techniques, but solving for higher roots or fractional roots requires the use of the mathematics that we will cover later. When solving for a square root of a complex number, we first set the square root equal to a general form of a complex number, then square both sides. By combining the real terms and combining the imaginary terms, we are able to form two equations and solve using substitution. For example: 4 3i a bi 4 3i a abi b i 4 3i (a b ) abi 4 a b and 3i abi 3 Solve for a: a b 3 Then substitute for a in the first equation: b 4 ( 3 b ) b 4 9 b 4b 16b 9 4b 4 4b 4 16b 9 0 b b 16 56 144 8 16 400 8 16 0 8 9, 1 009 Carnegie Learning, Inc. 134 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
Because b by definition is a real number, set the positive term equal to b : b 1 b 1 a 3 ( ) 4 3i 3 3 3 i, 3 i Check: ( 3 ( 3 i ) ( 3 ) ( 3 )( ) i ( i ) 18 4 4 3i ) i ( 3 ) ( 3 )( ) ( i ) i 18 4 1 4 i 4 i 4 3i 1 4 i 4 i 009 Carnegie Learning, Inc. Lesson.4 Operations with Complex Numbers 135
. Determine the square roots of 5 1i. Be prepared to share your work with another pair, group, or the entire class. 009 Carnegie Learning, Inc. 136 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
.5 Complex Roots and Zeros Complex Roots and Simplifying Complex Roots Objectives In this lesson, you will: Calculate complex roots of quadratic equations and complex zeros of quadratic functions. Interpret complex roots of quadratic equations and complex zeros of quadratic functions. Key Terms complex roots complex zeros Problem 1 1. Consider the function f(x) x 10x 37. a. Calculate the y-intercept. b. Calculate the equation for the axis of symmetry. c. Calculate the coordinates of the vertex. 009 Carnegie Learning, Inc. Lesson.5 Complex Roots and Simplifying Complex Roots 137
d. Calculate the x-intercept(s). e. Calculate the coordinates of the point on the parabola symmetric to the y-intercept. f. Sketch a graph of the function with parts (a) through (e) indicated on the graph. 009 Carnegie Learning, Inc.. Does the graph of f(x) x 10x 37 have x-intercepts? Were you able to calculate them? Why or why not? 138 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
3. Consider the function g(x) (x ) 1. a. Calculate the coordinates of the vertex. b. Calculate the equation for the axis of symmetry. c. Calculate the y-intercept. d. Calculate the coordinates of the point on the parabola symmetric to the y-intercept. e. Calculate the x-intercept(s). 009 Carnegie Learning, Inc. f. Sketch a graph of the function with parts (a) through (e) indicated on the graph. Lesson.5 Complex Roots and Simplifying Complex Roots 139
4. Does the graph of g(x) (x ) 1 have x-intercepts? Were you able to calculate them? Why or why not? 5. Solve each of the following quadratic equations for real solutions. a. 7x 4x 9 0 b. 10x 5x 4 0 009 Carnegie Learning, Inc. 6. Do either of the quadratic equations from Question 5 have real solutions? Why or why not? 140 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
Problem The quadratic equations and functions in Problem 1 had no real zeros or roots. Functions and equations that do not have real solutions may have complex roots or complex zeros. 1. Calculate the complex zeros for each quadratic function from Problem 1 Questions 1 and 3. Simplify completely and check your solutions. a. f(x) x 10x 37 009 Carnegie Learning, Inc. Lesson.5 Complex Roots and Simplifying Complex Roots 141
b. g(x) (x ) 1 009 Carnegie Learning, Inc. 14 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
. Calculate the complex roots for each quadratic equation from Problem 1 Question 5. Simplify completely and check your solutions. a. 7x 4 x 9 0 009 Carnegie Learning, Inc. Lesson.5 Complex Roots and Simplifying Complex Roots 143
b. 10x 5x 4 0 Be prepared to share your methods and solutions. 009 Carnegie Learning, Inc. 144 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
.6 Cases, Roots, and Graphs Solving Quadratic Inequalities Objectives In this lesson, you will: Solve quadratic inequalities using the case method. Solve quadratic inequalities using the roots method. Solve quadratic inequalities using the graphical method. Problem 1 The Case Method One method for solving quadratic inequalities is the case method. To use the case method, transform the inequality so that one side is zero, factor the quadratic expression, and examine the possible cases that satisfy the inequality. 1. If the product of two numbers is equal to zero, what do you know about the values of the two numbers?. If the product of two numbers is greater than zero, what do you know about the values of the two numbers? 009 Carnegie Learning, Inc. 3. If the product of two numbers is less than zero, what do you know about the values of the two numbers? Solve the inequality x 5x 7 1 using the case method by performing the following steps: x 5x 7 1 1 1 x 5x 6 0 (x 3)(x ) 0 Case 1: Both factors are positive (x 3) 0 and (x ) 0 x 3 and x 3 x Lesson.6 Solving Quadratic Inequalities 145
Case : Both factors are negative (x 3) 0 and (x ) 0 x 3 and x x Solution: x or x 3 4. Solve each quadratic inequality using the case method. a. 3x 7x 5x b. x 11x 10 009 Carnegie Learning, Inc. 146 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
Problem The Roots Method The case method works well for quadratic inequalities that can be factored. One method for solving quadratic inequalities that cannot be factored is the roots method. To use the roots method, transform the inequality so that one side is zero, calculate the roots of the corresponding quadratic equation, evaluate the equation for values on each interval defined by the roots, and examine the values that satisfy the inequality. Solve the inequality x x 6 0 using the roots method by performing the following steps: x x 6 0 a 1 b c 6 x b b 4ac a x () () 4(1)( 6) (1) x x 4 4 8 x 1 7 (4)(7) 7 ˇ x 3.646 or x 1.646ˇ The two roots divide the real numbers into three intervals: all numbers less than approximately 3.646, all numbers between approximately 3.646 and 1.646, and all numbers greater than approximately 1.646. Choose a value in each of these intervals. 009 Carnegie Learning, Inc. Try: 4, 0, ( 4) ( 4) 6 This is greater than 0. (0) (0) 6 6 This is not greater than 0. () () 6 This is greater than 0ˇ ˇ. The first and third intervals satisfy the inequality. Approximate solution: x 3.646 or x 1.646 Exact solution: x 1 7 or x 1 7 Lesson.6 Solving Quadratic Inequalities 147
1. Solve each quadratic inequality using the roots method. a. 3x 8x 6 009 Carnegie Learning, Inc. 148 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
b. x 5x 4 009 Carnegie Learning, Inc. Lesson.6 Solving Quadratic Inequalities 149
Problem 3 The Graphing Method A third method for solving quadratic inequalities is the graphing method. This method uses the characteristics of the graph. To use the graphing method, transform the inequality so that one side is zero, graph the corresponding quadratic function, calculate the x-intercepts of the parabola, and examine the portion of the parabola that satisfies the inequality. Solve the inequality x 5x 7 1 using the graphing method by performing the following steps. x 5x 7 1 x 5x 6 0 Corresponding quadratic function: y x 5x 6 Graph the function y x 5x 6. y 8 6 4 8 6 4 4 6 8 x 4 6 8 Calculate the x-intercepts of the parabola. x 5x 6 0 (x 3)(x ) 0 x, 3 (,0), (3,0) 009 Carnegie Learning, Inc. Plot the x-intercepts. The portion of the parabola to the left and right of the x-intercepts has y-values that are greater than zero. Solution: x or x 3 150 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
1. Solve x 5x 7 1 using the graphing method.. If you know the x-intercepts of the function, do you have to graph the function to find a solution? Explain. 3. Solve each quadratic inequality using the graphing method. a. Solve 3x 10x 3. 009 Carnegie Learning, Inc. Lesson.6 Solving Quadratic Inequalities 151
b. Solve x 5x 5. Be prepared to share your methods and solutions. 009 Carnegie Learning, Inc. 15 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
.7 Carl Freidrich Gauss: Child Prodigy Arithmetic Sequences, Series, and Their Partial Sums Objectives In this lesson, you will: Write arithmetic sequences. Calculate the next terms of a sequence. Define arithmetic sequences using explicit formulas. Define arithmetic sequences using recursive formulas. Calculate specific terms of an arithmetic series. Determine the sum of finite arithmetic series. Key Terms sequence term arithmetic sequence index explicit formula recursive formula series arithmetic series finite series infinite series A sequence is an ordered list of numbers. Each number in the sequence is called a term. The domain of a sequence is consecutive integers usually beginning with 1 and ending with the number of terms in the sequence. The range of a sequence is all values of the terms of the sequence. 009 Carnegie Learning, Inc. Problem 1 1. For each sequence, list the next three terms of the sequence. Then describe how to generate each new term. a. 5, 9, 13, 17 b. 1, 3, 5, 7, 9 c. 1,.5, 4, 5.5 d. 1, 9, 6, 3, 0 Lesson.7 Arithmetic Sequences, Series, and Their Partial Sums 153
. What do you notice about the process you used to generate new terms for each sequence in Question 1? Each sequence in Question 1 is an arithmetic sequence. An arithmetic sequence is a sequence of numbers that has a common difference, d, between each two successive terms. The terms of a sequence are referenced by a n where n, called the index, is the position of the term in the sequence. For example, for the sequence 5, 9, 13, 17, a 1 5, a 9, a 3 13, and a 4 17. 3. The first term of a sequence is. The difference between successive terms is 3. Write the first six terms of the sequence. 4. The first term of a sequence is 0. The difference between successive terms is 0.5. Write the first eight terms of the sequence. 5. The first term of a sequence is 1. The difference between successive terms is. Write the first five terms of the sequence. Problem 1. What is the common difference, d, of the sequence 5, 7, 9, 11, 13, 15, 17, 19, 1,?. What is the first term of the sequence, a 1? 009 Carnegie Learning, Inc. 3. What is a? How can you calculate this value from the first term? 154 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
4. Complete the table of values representing the sequence. Term Number 1 Term 5 5. The sequence can be represented by a linear function. Why? 6. Write a linear function to represent the sequence. 7. What does the slope of the linear function mean in terms of the sequence? 8. What does the y-intercept of the linear function mean in terms of the sequence? 9. What is a 4? How could you calculate a 4 directly from f(x)? 10. What is a 10? How could you calculate a 10 directly from f(x)? 009 Carnegie Learning, Inc. 11. What is a 50? How could you calculate a 50 directly from f(x)? A rule, such as f(x) x 3, for calculating a particular term in a sequence is called an explicit formula. Explicit formulas are useful for calculating remote terms in a sequence, such as a 50, without calculating all the terms that come before it. 1. Let a n represent the nth term. Write a rule to calculate a n. 13. Let a n 1 represent the term before the nth term. Write a rule to calculate a n 1. Lesson.7 Arithmetic Sequences, Series, and Their Partial Sums 155
14. The rule for an can also be written as an (n 1) 5. How could you generate this rule from the original sequence? 15. Show that the two rules for an are equivalent. 16. Which rule for an is easier to generate for a given arithmetic sequence? Why? Problem 3 Sequences can also be defined using a recursive formula. A recursive formula expresses each new term based on the preceding term. The sequence 5, 7, 9, 11, 13, 15, 17, 19, 1 can be defined recursively as an an 1 with a1 5. Consider the arithmetic sequence 4, 6, 8, 10,... 1. What is the common difference, d? How can you use the common difference to calculate each next term? 3. Use the recursive formula to calculate a5, a6, and a7. 4. Can you use the recursive formula to calculate a0? Explain 5. Write an explicit formula to calculate an. 156 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities 009 Carnegie Learning, Inc.. Write a recursive formula to calculate an.
6. Use the explicit formula to calculate a 0. 7. Summarize how to write an explicit and a recursive formula for an arithmetic sequence. Problem 4 Calculate the first six terms of each sequence defined by an explicit formula. Then write a recursive formula for the sequence. 1. a n 5n 3. a n 3n 4 3. a n 6(n 1) 5 009 Carnegie Learning, Inc. 4. a n (n 1) 5 Lesson.7 Arithmetic Sequences, Series, and Their Partial Sums 157
Problem 5 Write a recursive and an explicit formula to calculate an for each sequence. 1. 0, 15, 10, 5, 0,., 18, 34, 50, 66, 3. 10, 14, 18,, 4. 100, 7, 44, 16, Problem 6 The sum of the first 100 natural numbers is a finite series and can be written as S 1 3 4 99 100. The sum of the natural numbers is an infinite series and can be written as S 1 3 4 In general, Sn is the sum of the first n terms of a sequence. For example, S is the sum of the first two terms and S10 is the sum of the first 10 terms. 1. Calculate each series. a. 5 9 13 17 S b. 1 3 5 7 9 S 158 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities 009 Carnegie Learning, Inc. A series is the sum of a sequence of numbers. An arithmetic series is the sum of an arithmetic sequence. A finite series is the sum of a finite number of terms. An infinite series is the sum of an infinite number of terms.
. Calculate each indicated sum. a. 5 9 13 17 1 S S 4 S b. 1 9 6 3 0 ( 3) ( 6) ( 9) S S 6 S A series is sometimes written using sigma notation. Sigma is a Greek letter and is used to indicate a sum. S n n a i a 1 a... a n i 1 The sigma notation is read as the sum of all terms a i for i from 1 to n. The sigma notation can also be written using the explicit formula for the sequence. For example, consider the sequence 5, 11, 17, 3, 9, 35, An explicit formula for any term a n of the sequence is a n 5 6(n 1). n S n a i 5 6(i 1) i 1 n i 1 3. Use the explicit formula for the sequence to write a formula for each series using sigma notation. 009 Carnegie Learning, Inc. a. 5 9 13 17 1 b. 1 9 6 3 0 ( 3) ( 6) ( 9) Lesson.7 Arithmetic Sequences, Series, and Their Partial Sums 159
Problem 7 Calculating an arithmetic series with 100 terms, such as 4 6 198 00, is very time consuming. There must be an easier way to calculate the sum! Although there is some debate about the truth of this story, this same problem was supposedly given to famous mathematician Carl Friedrich Gauss in primary school. At this young age, he was able to come up with an easier method to solve the problem, which is known as Gauss s Solution. To use Gauss s Solution, write any arithmetic series in ascending order. Then write the same series in descending order below it, making sure to align the terms. The series 4 6 198 00 can be written as 4 6 198 00 00 198 196 4 Next calculate the sum of each pair of vertical terms. 1. What do you notice about the sum of each pair of vertical terms?. What is the sum of each pair of vertical terms? 3. How many pairs of vertical terms are there? 4. What is the sum of all the pairs of vertical terms? 5. The sum in Question 4 includes each of the terms of the original series twice. What is the sum of the original series? 009 Carnegie Learning, Inc. 6. What do you notice about the terms of the sequence and the sum of each vertical pair? 160 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
7. Given an arithmetic series with n terms, describe how to calculate the sum of the arithmetic series. 8. Calculate the sum of the arithmetic series 1 3 5 7 9 11 13 15 17 19. Problem 8 In Problem 7, we learned that: The number of vertical pairs is equal to the number of terms in the series. The sum of each vertical pair is equal to the sum of the first and last terms of the series. The sum of the arithmetic series is equal to half the product of these two quantities. For an arithmetic series with n terms, the sum of the first n terms is or S n n(a 1 a n ) S n n (a 1 a n ) where a 1 is the first term in the series and a n is the last term of the series. Calculate each sum. 1. 5 9 13 17 1 009 Carnegie Learning, Inc. S S 3. 1 9 6 3 0 ( 3) ( 6) ( 9) S S 7 Lesson.7 Arithmetic Sequences, Series, and Their Partial Sums 161
3. 5 9 13 17 1 5 49 53 S Be prepared to share your solutions and explanations with your class. 009 Carnegie Learning, Inc. 16 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
.8 Mathematics Empowers Us To Do Superhuman Mental Calculations! Modeling Partial Sums of Arithmetic Series with Quadratic Equations Objectives In this lesson, you will: Write a function to represent the sum of the first n terms of an arithmetic sequence. Represent geometric figures using arithmetic sequences and series. Problem 1 Adding Odd Numbers Emma claims that she can calculate the sum of the first 0 odd numbers in her head in one second! Sound amazing? Let s learn her secret! Consider the arithmetic sequence of odd numbers 1, 3, 5, 7, with a1 1. 009 Carnegie Learning, Inc. 1. Write an explicit formula to calculate any term of the sequence. Write the formula in simplest terms.. Use Gauss s Solution to calculate the sum of the first 0 odd numbers. To use Gauss s Solution, you have to know or calculate the first and last terms of the series. It is possible to calculate the sum of a series without having to calculate the last term. 3. Substitute the known value of a1 and the algebraic expression for an into the n(a1 an ) formula Sn and simplify completely. Lesson.8 Modeling Partial Sums of Arithmetic Series with Quadratic Equations 163
4. The sum of the first n odd integers can be written as a function. What kind of function is it? 5. Calculate the sum of the first 5 odd integers by adding the terms or using Gauss s Solution. 6. Calculate the sum of the first 5 odd integers using the formula from Question 3. 7. Compare the answers from Questions 5 and 6. What do you notice? 8. Calculate the sum of the first 0 odd integers using the formula from Question 3. 9. Compare the answers from Questions and 8. What do you notice? Problem Adding Even Numbers David claims that he can calculate the sum of the first 0 even numbers the same way Emma calculated the sum of the first 0 odd numbers. Is he correct? Consider the arithmetic sequence of even numbers, 4, 6, 8, with a 1. 1. Write an explicit formula to calculate any term of the sequence. Write the formula in simplest terms.. Use Gauss s Solution to calculate the sum of the first 0 even numbers. 009 Carnegie Learning, Inc. 3. Substitute the known value of a 1 and the algebraic expression for a n into the formula S n n(a 1 a n ) and simplify completely. 164 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
4. The sum of the first n even integers can be written as a function. What kind of function is it? 5. Calculate the sum of the first 5 even integers by adding the terms or using Gauss Solution. 6. Calculate the sum of the first 5 even integers using the formula from Question 3. 7. Compare the answers from Questions 5 and 6. What do you notice? 8. Calculate the sum of the first 0 even integers using the formula from Question 3. 9. Compare the answers from Questions and 8. What do you notice? 10. Remember David s original claim that he could calculate the sum of the first 0 even numbers the same way Emma calculated the sum of the first 0 odd numbers. Was David correct? Explain. 009 Carnegie Learning, Inc. Lesson.8 Modeling Partial Sums of Arithmetic Series with Quadratic Equations 165
Problem 3 Consider the arithmetic sequence 1, 5, 9, 13, where a1 1. 1. Write an explicit formula to calculate any term of the sequence. Write the formula in simplest terms.. Substitute the known value of a1 and the algebraic expression for an into the n(a1 an ) formula Sn and simplify completely. 3. The sum of the first n terms can be written as a function. What kind of function is it? 4. Calculate the sum of the first 100 terms of the sequence. Problem 4 Graph Paper Drawings Sam was drawing on graph paper. The first three figures he drew are shown. He continued this pattern to 100 figures. Figure Figure 3 1. Sketch Figure 4 and Figure 5.. Use the first 5 figures to write a sequence that represents the number of blocks used in each figure. 3. Write an explicit formula to calculate any term of the sequence. Write the formula in simplest terms. 166 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities 009 Carnegie Learning, Inc. Figure 1
4. How does the formula relate to the diagram? 5. Substitute the known value of a 1 and the algebraic expression for a n into the formula S n n(a 1 a n ) and simplify completely. 6. The sum of the first n terms can be written as a function. What kind of function is it? 7. Calculate the total number of blocks used in the first 5 figures by adding the terms or using Gauss s Solution. 8. Calculate the total number of blocks used in the first 5 figures using the formula from Question 5. 9. Compare the answers from Questions 7 and 8. What do you notice? 009 Carnegie Learning, Inc. 10. Calculate the total number of blocks used in the first 0 figures. Lesson.8 Modeling Partial Sums of Arithmetic Series with Quadratic Equations 167
Problem 5 Honey Cells Honey bees build hexagonal honey cells from a single cell, such as the one shown in Figure 1. Layers of honey cells are then built around the edges as shown in Figures and 3. Figure 1 Figure Figure 3 1. Sketch what the fourth and fifth figures would look like.. Use the first 5 figures to write a sequence that represents the number of cells used in each layer. The cell in Figure 1 does not represent a layer. 3. Write an explicit formula to calculate any term of the sequence. Write the formula in simplest terms. 009 Carnegie Learning, Inc. 4. How does the formula relate to the diagram? 168 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
5. Substitute the known value of a 1 and the algebraic expression for a n into the formula S n n(a 1 a n ) and simplify completely. 6. The sum of the first n terms can be written as a function. What kind of function is it? 7. Calculate the total number of cells used in the first 5 layers by adding the terms or using Gauss s Solution. 8. Calculate the total number of cells used in the first 5 layers using the formula from Question 5. 9. Compare the answers from Questions 7 and 8. What do you notice? 10. Calculate the total number of cells used in the first 0 layers. 009 Carnegie Learning, Inc. Lesson.8 Modeling Partial Sums of Arithmetic Series with Quadratic Equations 169
Problem 6 Tiles Using the first three figures in this tile pattern, answer the following questions. Figure 1 Figure Figure 3 1. Sketch Figure 4 and Figure 5.. Use the first 5 figures to write a sequence that represents the number of blocks used in each figure. 3. Is this an arithmetic sequence? Explain. 4. Write an explicit formula to calculate the number of blocks used in any figure of the sequence. Write the formula in simplest terms. 5. How is this formula represented in the diagram? 6. Calculate the total number of tiles used in the 50th figure. Be prepared to share your solutions and explanations with your class. 009 Carnegie Learning, Inc. 170 Chapter The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities