Algebra I Quadratics

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Algebra I Quadratics 2015-11-04 www.njctl.org 2

Key Terms Table of Contents Click on the topic to go to that section Characteristics of Quadratic Equations Transforming Quadratic Equations Graphing Quadratic Equations Solve Quadratic Equations by Graphing Solve Quadratic Equations by Factoring Solve Quadratic Equations Using Square Roots Solve Quadratic Equations by Completing the Square The Discriminant Solve Quadratic Equations by Using the Quadratic Formula Solving Application Problems 3

Key Terms Return to Table of Contents 4

Axis of Symmetry Axis of symmetry: The vertical line that divides a parabola into two symmetrical halves 5

Parabolas Maximum: The y-value of the vertex if a < 0 and the parabola opens downward Minimum: The y-value of the vertex if a > 0 and the parabola opens upward (+ a) Max Min (- a) Parabola: The curve result of graphing a quadratic equation 6

Quadratics Quadratic Equation: An equation that can be written in the standard form ax 2 + bx + c = 0. Where a, b and c are real numbers and a does not = 0. Vertex: The highest or lowest point on a parabola. Zero of a Function: An x value that makes the function equal zero. 7

Characteristics of Quadratic Equations Return to Table of Contents 8

Quadratics A quadratic equation is an equation of the form ax 2 + bx + c = 0, where a is not equal to 0. The form ax 2 + bx + c = 0 is called the standard form of the quadratic equation. The standard form is not unique. For example, x 2 - x + 1 = 0 can be written as the equivalent equation -x 2 + x - 1 = 0. Also, 4x 2-2x + 2 = 0 can be written as the equivalent equation 2x 2 - x + 1 = 0. Why is this equivalent? 9

Writing Quadratic Equations Practice writing quadratic equations in standard form: (Simplify if possible.) Write 2x 2 = x + 4 in standard form: Answer 10

1 Write 3x = -x 2 + 7 in standard form: A. x 2 + 3x-7= 0 B. x 2-3x +7=0 C. -x 2-3x -7= 0 Answer 11

2 Write 6x 2-6x = 12 in standard form: A. 6x 2-6x -12 = 0 B. x 2 - x - 2 = 0 C. -x 2 + x + 2 = 0 Answer 12

3 Write 3x - 2 = 5x in standard form: A. 2x + 2 = 0 B. -2x - 2 = 0 C. not a quadratic equation Answer 13

Characteristics of Quadratic Functions The graph of a quadratic is a parabola, a u-shaped figure. The parabola will open upward or downward. downward upward 14

Characteristics of Quadratic Functions A parabola that opens upward contains a vertex that is a minimum point. A parabola that opens downward contains a vertex that is a maximum point. vertex vertex 15

Characteristics of Quadratic Functions The domain of a quadratic function is all real numbers. D = Reals 16

Characteristics of Quadratic Functions To determine the range of a quadratic function, ask yourself two questions: > Is the vertex a minimum or maximum? > What is the y-value of the vertex? If the vertex is a minimum, then the range is all real numbers greater than or equal to the y-value. The range of this quadratic is [ 6, ) 17

Characteristics of Quadratic Functions If the vertex is a maximum, then the range is all real numbers less than or equal to the y-value. The range of this quadratic is (,10] 18

Characteristics of Quadratic Functions 7. An axis of symmetry (also known as a line of symmetry) will divide the parabola into mirror images. The line of symmetry is always a vertical line of the form x = b 2a y = 2x 2 8x + 2 x = ( 8) 2(2) = 2 x=2 Teacher Notes 19

Characteristics of Quadratic Functions To find the axis of symmetry simply plug the values of a and b into the equation: Remember the form ax 2 + bx + c. In this example a = 2, b = -8 and c =2 a b c x = b 2a x=2 y = 2x 2 8x + 2 x = ( 8) 2(2) = 2 20

Characteristics of Quadratic Functions The x-intercepts are the points at which a parabola intersects the x-axis. These points are also known as zeroes, roots or solutions and solution sets. Each quadratic equation will have two, one or no real x-intercepts. 21

4 The vertical line that divides a parabola into two symmetrical halves is called... A discriminant B perfect square C axis of symmetry Answer D vertex E slice 22

5 What are the vertex and axis of symmetry of the parabola shown in the diagram below? A B vertex: (1, 4); axis of symmetry: x = 1 vertex: (1, 4); axis of symmetry: x = 4 C D vertex: ( 4,1); axis of symmetry: x = 1 vertex: ( 4,1); axis of symmetry: x = 4 Answer From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011. 23

6 The equation y = x 2 + 3x 18 is graphed on the set of axes below. Based on this graph, what are the roots of the equation x 2 + 3x 18 = 0? A B C D 3 and 6 0 and 18 3 and 6 3 and 18 Answer From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011. 24

7 The equation y = x 2 2x + 8 is graphed on the set of axes below. Based on this graph, what are the roots of the equation x 2 2x + 8 = 0? A 8 and 0 B 2 and 4 C 9 and 1 D 4 and 2 Answer From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011. 25

8 What is an equation of the axis of symmetry of the parabola represented by y = x 2 + 6x 4? A x = 3 B y = 3 C x = 6 D y = 6 Answer From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011. 26

9 The height, y, of a ball tossed into the air can be represented by the equation y = x 2 + 10x + 3, where x is the elapsed time. What is the equation of the axis of symmetry of this parabola? A y = 5 B y = 5 C x = 5 D x = 5 Answer From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011. 27

10 What is the equation of the axis of symmetry of the parabola shown in the diagram below? A x = 0.5 B x = 2 C x = 4.5 D x = 13 18 16 14 12 10 8 6 4 2 5 6 7 8 9 10 Answer From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011. 28

Transforming Quadratic Equations Return to Table of Contents 29

Quadratic Parent Equation y = 2 x 2 3 The quadratic parent equation is y = x 2. The graph of all other quadratic equations are transformations of the graph of y= x 2. x x 2 y = x 2-3 9-2 4-1 1 0 0 1 1 2 4 3 9 30

Quadratic Parent Equation The quadratic parent equation is y = x 2. How is y = x 2 changed into y = 2x 2? y = 2x 2 x 2 y = x 2-3 18-2 8-1 2 0 0 1 2 2 8 3 18 31

Quadratic Parent Equation The quadratic parent equation is y = x 2. How is y = x 2 changed into y =.5x 2? y = x 2 y = x 2 1 2 x 0.5-3 4.5-2 2-1 0.5 0 0 1 0.5 2 2 3 4.5 32

What Does "A" Do? What does "a" do in y = ax 2 + bx + c? How does a > 0 affect the parabola? How does a < 0 affect the parabola? y = x 2 y = x 2 33

What Does "A" Do? What does "a" also do in y =ax 2 + bx +c? y = How does your conclusion about "a" change as "a" changes? 1 2 x2 y = 3x 2 y = x 2 y = 1x 2 y = 3x 2 y = x 2 1 2 34

What Does "A" Do? What does "a" do in y = ax 2 + bx + c? If a > 0, the graph opens up. If a < 0, the graph opens down. If the absolute value of a is > 1, then the graph of the equation is narrower than the graph of the parent equation. If the absolute value of a is < 1, then the graph of the equation is wider than the graph of the parent equation. 35

11 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent equation. A B up, wider up, narrower y =.3x 2 Answer C down, wider D down, narrower 36

12 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent function. y = 4x 2 A B C D up, wider up, narrower down, wider down, narrower Answer 37

13 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent equation. A B C D y = 2x 2 + 100x + 45 up, wider up, narrower down, wider down, narrower Answer 38

14 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent function. A B up, wider up, narrower y = x 2 2 3 Answer C down, wider D down, narrower 39

15 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent function. A B up, wider up, narrower y = x 2 7 5 Answer C down, wider D down, narrower 40

What Does "C" Do? What does "c" do in y = ax 2 + bx + c? y = x 2 + 6 y = x 2 + 3 y = x 2 y = x 2 2 y = x 2 5 y = x 2 9 41

What Does "C" Do? What does "c" do in y = ax 2 + bx + c? "c" moves the graph up or down the same value as "c." "c" is the y- intercept. 42

16 Without graphing, what is the y- intercept of the the given parabola? y = x 2 + 17 Answer 43

17 Without graphing, what is the y- intercept of the the given parabola? y = x 2 6 Answer 44

18 Without graphing, what is the y- intercept of the the given parabola? y = 3x 2 + 13x 9 Answer 45

19 Without graphing, what is the y- intercept of the the given parabola? y = 2x 2 + 5x Answer 46

20 Choose all that apply to the following quadratic: f(x) =.7x 2 4 A opens up A y-intercept of y = 4 B C opens down wider than parent function B C y-intercept of y = 2 y-intercept of y = 0 Answer D narrower than parent D y-intercept of y = 2 function E y-intercept of y = 4 F y-intercept of y = 6 47

21 Choose all that apply to the following quadratic: A B C D opens up opens down wider than parent function narrower than parent function f(x) = 4 x 2 6x 3 E y-intercept of y = 4 F y-intercept of y = 2 G y-intercept of y = 0 H y-intercept of y = 2 I y-intercept of y = 4 J y-intercept of y = 6 Answer 48

22 The diagram below shows the graph of y = x 2 c. Which diagram shows the graph of y = x 2 c? Answer A B C D From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011. 49

Graphing Quadratic Equations Return to Table of Contents 50

Graph by Following Six Steps: Step 1 - Find Axis of Symmetry Step 2 - Find Vertex Step 3 - Find Y intercept Step 4 - Find two more points Step 5 - Partially graph Step 6 - Reflect 51

Step 1 - Find Axis of Symmetry Axis of Symmetry What is the Axis of Symmetry? Axis of Symmetry Teacher Notes 52

Step 1 - Find Axis of Symmetry Formula: a = 3 b = 6 x = b 2a x = ( 6) = 6 = 1 2(3) 6 Graph y = 3x 2 6x + 1 The axis of symmetry is x = 1. 53

Step 2 - Find Vertex Step 2 - Find the vertex by substituting the value of x (the axis of symmetry) into the equation to get y. y = 3x 2 6x + 1 a = 3, b = 6 and c = 1 y = 3(1) 2 + 6(1) + 1 y = 3 6 + 1 y = 2 Vertex = (1, 2) 54

Step 3 - Find y intercept What is the y-intercept? y- intercept Teacher Notes 55

Step 3 - Find y intercept Graph y = 3x 2 6x + 1 The y- intercept is always the c value, because x = 0. y = ax 2 + bx + c y = 3x 2 6x + 1 c = 1 The y-intercept is 1 and the graph passes through (0,1). 56

Step 4 - Find Two More Points Graph y = 3x 2 6x + 1 Find two more points on the parabola. Choose different values of x and plug in to find points. Let's pick x = 1 and x = 2 y = 3x 2 6x + 1 y = 3( 1) 2 6( 1) + 1 y = 3 + 6 + 1 y = 10 ( 1,10) 57

Step 4 - Find Two More Points (continued) Graph y = 3x 2 6x + 1 y = 3x 2 6x + 1 y = 3( 2) 2 6( 2) + 1 y = 3(4) + 12 + 1 y = 25 ( 2, 25) 58

Step 5 - Graph the Axis of Symmetry Graph the axis of symmetry, the vertex, the point containing the y-intercept and two other points. 59

Step 6 - Reflect the Points Reflect the points across the axis of symmetry. Connect the points with a smooth curve. (4,25) 60

23 What is the axis of symmetry for y = x 2 + 2x - 3 (Step 1)? Answer 61

24 What is the vertex for y = x 2 + 2x - 3 (Step 2)? A (-1, -4) B (1, -4) C (-1, 4) Answer 62

25 What is the y-intercept for y = x 2 + 2x - 3 (Step 3)? A -3 B 3 Answer 63

Graph y= x 2 + 2x 3 axis of symmetry = 1 vertex = 1, 4 y intercept = 3 2 other points (step 4) (1,0) (2,5) Partially graph (step 5) Reflect (step 6) 64

Graph y = 2x 2 6x + 4 65

Graph y = x 2 4x + 5 66

Graph y = 3x 2 7 67

Solve Quadratic Equations by Graphing Return to Table of Contents 68

Find the Zeros One way to solve a quadratic equation in standard form is find the zeros by graphing. A zero is the point at which the parabola intersects the x-axis. A quadratic may have one, two or no zeros. 69

Find the Zeros How many zeros do the parabolas have? What are the values of the zeros? click No zeroes (doesn't cross the "x" axis) click 2 zeroes; x = -1 and x=3 click 1 zero; x=1 70

Review To solve a quadratic equation by graphing follow the 6 step process we already learned. Step 1 - Find Axis of Symmetry Step 2 - Find Vertex Step 3 - Find Y intercept Step 4 - Find two more points Step 5 - Partially graph Step 6 - Reflect 71

26 Solve the equation by graphing. 12x + 18 = 2x 2 Which of these is in standard form? A B C y = 2x 2 12x + 18 y = 2x 2 12x + 18 y = 2x 2 + 12x 18 Answer 72

27 What is the axis of symmetry? y = 2x 2 + 12x 18 A 3 B 3 C 4 D 5 Answer 73

28 y = 2x 2 + 12x 18 What is the vertex? A (3,0) B ( 3,0) C (4,0) D ( 5,0) Answer 74

29 y = 2x 2 + 12x 18 What is the y- intercept? A (0, 0) B (0, 18) C (0, 18) Answer D (0, 12) 75

30 A If two other points are (5, 8) and (4, 2),what does the graph of y = 2x 2 + 12x 18 look like? B Answer C D 76

31 y = 2x 2 + 12x 18 What is(are) the zero(s)? A 18 B 4 C 3 D 8 click for graph of answer Answer 77

Solve Quadratic Equations by Factoring Return to Table of Contents 78

Solving Quadratic Equations by Factoring Review of factoring - To factor a quadratic trinomial of the form x 2 + bx + c, find two factors of c whose sum is b. Example - To factor x 2 + 9x + 18, look for factors whose sum is 9. Factors of 18 Sum 1 and 18 19 2 and 9 11 3 and 6 9 x 2 + 9x + 18 = (x + 3)(x + 6) 79

Solving Quadratic Equations by Factoring When c is positive, it's factors have the same sign. The sign of b tells you whether the factors are positive or negative. When b is positive, the factors are positive. When b is negative, the factors are negative. 80

Solving Quadratic Equations by Factoring Remember the FOIL method for multiplying binomials 1. Multiply the First terms (x + 3)(x + 2) x x = x 2 2. Multiply the Outer terms (x + 3)(x + 2) x 2 = 2x 3. Multiply the Inner terms (x + 3)(x + 2) 3 x = 3x 4. Multiply the Last terms (x + 3)(x + 2) 3 2 = 6 (x + 3)(x + 2) = x 2 + 2x + 3x + 6 = x 2 + 5x + 6 F O I L 81

Zero Product Property For all real numbers a and b, if the product of two quantities equals zero, at least one of the quantities equals zero. Numbers Algebra 3(0) = 0 If ab = 0, 4(0) = 0 Then a = 0 or b = 0 82

Zero Product Property Example 1: Solve x 2 + 4x 12 = 0 (x + 6) (x 2) = 0 Use "FUSE"! Factor the trinomial using the FOIL method. x + 6 = 0 or x 2 = 0 6 6 + 2 +2 x = 6 x = 2 6 2 + 4( 6) 12 = 0 6 2 + ( 24) 12 = 0 36 24 12 = 0 0 = 0 or 2 2 + 4(2) 12 = 0 4 + 8 12 = 0 0 = 0 Use the Zero property Substitue found value into original equation Equal - problem solved! The solutions are -6 and 2. 83

Zero Product Property Example 2: Solve x 2 + 36 = 12x 12x 12x x 2 12x + 36 = 0 (x 6)(x 6) = 0 x 6 = 0 +6 +6 x = 6 6 2 + 36 = 12(6) 36 + 36 = 72 The equation has to be written in standard form (ax 2 + bx + c). So subtract 12x from both sides. Factor the trinomial using the FOIL method. Use the Zero property Substitue found value into original equation 72 = 72 Equal - problem solved! 84

Zero Product Property Example 3: Solve x 2 16x + 48= 0 (x 4)(x 12) = 0 x 4 = 0 x 12 = 0 +4 +4 +12 +12 x = 4 x = 12 4 2 16(4) + 48 = 0 16 64 + 48 = 0 48+48 = 0 0 = 0 Factor the trinomial using the FOIL method. Use the Zero property Substitue found value into original equation 12 2 16(12) + 48 = 0 144 192 + 48 = 0 48 + 48 = 0 0 = 0 48 Equal - problem solved! 85

32 Solve x 2 5x + 6 = 0 A 7 B 5 C 3 D 2 E 2 F 3 G 5 H 6 I 7 J 15 Answer 86

33 Solve m 2 + 10m + 25 = 0 A 7 B 5 C 3 D 2 E 2 F 3 G 5 H 6 I 7 J 15 Answer 87

34 Solve h 2 h = 12 A 12 B 4 C 3 D 2 E 2 F 3 G 4 H 6 I 8 J 12 Answer 88

35 Solve d 2 35d = 2d A 7 B 5 C 3 D 35 E 12 F 0 G 5 H 6 I 7 J 37 Answer 89

36 Solve 8y 2 + 2y = 3 A 3 / 4 B 1 / 2 C 4 / 3 D 2 E 2 F 3 / 4 G 1 / 2 H 4 / 3 I 3 J 3 Answer 90

37 Which equation has roots of 3 and 5? A x 2 + 2x 15 = 0 B x 2 2x 15 = 0 C x 2 + 2x + 15 = 0 D x 2 2x + 15 = 0 Answer 91

Solve Quadratic Equations Using Square Roots Return to Table of Contents 92

Square Root Method You can solve a quadratic equation by the square root method if you can write it in the form: x² = c If x and c are algebraic expressions, then: x = c or x = c written as: x = ± c 93

Square Root Method Solve for z: z² = 49 z = ± 49 z = ±7 The solution set is 7 and 7 94

Square Root Method A quadratic equation of the form x 2 = c can be solved using the Square Root Property. Example: Solve 4x 2 = 20 4x 2 = 20 4 4 x 2 = 5 Divide both sides by 4 to isolate x² The solution set is 5 and 5 x = ± 5 95

Square Root Method Solve 5x² = 20 using the square root method: 5x 2 = 20 5 5 x 2 = 4 x = 4 or x = 4 x = ± 2 96

Square Root Method Solve (2x 1)² = 20 using the square root method. click 2x 1 = 20 2x 1 = (4)(5) 2x 1 = 2 5 2x = 1 + 2 5 1 + 2 5 x = 2 or click 2x 1 = 20 2x 1 = (4)(5) 2x 1 = 2 5 2x = 1 2 5 1 2 5 x = 2 solution: x = 1 ± 2 5 click 2 97

38 When you take the square root of a real number, your answer will always be positive. True False Answer 98

39 If x 2 = 16, then x = A 4 B 2 C 2 D 26 E 4 Answer 99

40 If y 2 = 4, then y = A 4 B 2 C 2 D 26 E 4 Answer 100

41 If 8j 2 = 96, then j = A 3 2 B 2 3 C 2 3 D 3 2 E ±12 Answer 101

42 If 4h 2 10= 30, then h = A 10 B 2 5 C 2 5 D 10 E ±10 Answer 102

43 If (3g 9) 2 + 7= 43, then g = A 1 B 9 5 2 3 C 9 + 5 2 3 D 5 Answer E ±3 103

Solving Quadratic Equations by Completing the Square Return to Table of Contents 104

Find the Missing Value of "C" Before we can solve the quadratic equation, we first have to find the missing value of C. To do this, simply take the value of b, divide it in 2 and then square the result. ax 2 +bx+c (b/2) 2 Find the value that completes the square. 8/2 = 4 4 2 = 16 x 2 + 8x + x 2 + 20x + 100 x 2 16x + 64 x 2 2x + 1 105

44 Find ( b /2) 2 if b = 14 Answer 106

45 Find ( b /2) 2 if b = 12 Answer 107

46 Complete the square to form a perfect square t rinomial x 2 + 18x +? Answer 108

47 Complete the square to form a perfect square trinomial x 2 6x +? Answer 109

Solving Quadratic Equations by Completing the Square Step 1 - Write the equation in the form x 2 + bx = c Step 2 - Find (b 2) 2 Step 3 - Complete the square by adding (b 2) 2 to both sides of the equation. Step 4 - Factor the perfect square trinomial. Step 5 - Take the square root of both sides Step 6 - Write two equations, using both the positive and negative square root and solve each equation. 110

Solving Quadratic Equations by Completing the Square Let's look at an example to solve: x 2 + 14x = 15 x 2 + 14x = 15 Step 1 - Already done! (14 2) 2 = 49 Step 2 - Find (b 2) 2 x 2 + 14x + 49 = 15 + 49 Step 3 - Add 49 to both sides (x + 7) 2 = 64 Step 4 - Factor and simplify x + 7 = ±8 Step 5 - Take the square root of both sides x + 7 = 8 or x + 7 = 8 Step 6 - Write and solve two equations x = 1 or x = 15 111

Solving Quadratic Equations by Completing the Square Another example to solve: x 2 2x 2 = 0 x 2 2x 2 = 0 Step 1 - Write as x 2 +bx=c +2 +2 x 2 2x = 2 ( 2 2) 2 = ( 1) 2 = 1 Step 2 - Find (b 2) 2 x 2 2x + 1 = 2 + 1 Step 3 - Add 1 to both sides (x 1) 2 = 3 Step 4 - Factor and simplify x 1 = ± 3 x 1 = 3 or x 1 = 3 x = 1 + 3 or x = 1 3 Step 5 - Take the square root of both sides Step 6 - Write and solve two equations 112

48 Solve the following by completing the square : x 2 + 6x = 5 A 5 B 2 C 1 D 5 E 2 Answer 113

49 Solve the following by completing the square: x 2 8x = 20 A 10 B 2 C 1 D 10 E 2 Answer 114

50 Solve the following by completing the square : 36x = 3x 2 + 108 A 6 B 6 C 0 D 6 E 6 Answer 115

Solve A more difficult example: 3x 2 10x = 3 3 3 3 10x x 2 = 1 3 ( ) ( ) ( ) ( 5 4 x = ± 3 3 2 2 2 3x 2 10x = 3 10 3 2 = 10 x 1 = 5 = 25 3 2 3 9 10x 25 25 x 2 + = 1 + 3 9 9 2 ) 5 16 x = 3 9 Write as x 2 +bx=c Find (b 2) 2 Add 25/9 to both sides Factor and simplify Take the square root of both sides Answer 116

51 Solve the following by completing the square: 4x 2 7x 2 = 0 A 1 4 B 5 4 2 Answer C 1 4 D 5 4 2 E 2 117

The Discriminant Return to Table of Contents 118

The Discriminant Discriminant - the part of the equation under the radical sign in a quadratic equation. x = b ± b 2 4ac 2a b 2 4ac is the discriminant 119

The Discriminant ax 2 + bx + c = 0 The discriminant, b 2 4ac, or the part of the equation under the radical sign, may be used to determine the number of real solutions there are to a quadratic equation. If b 2 4ac > 0, the equation has two real solutions If b 2 4ac = 0, the equation has one real solution If b 2 4ac < 0, the equation has no real solutions 120

The Discriminant Remember: The square root of a positive number has two solutions. The square root of zero is 0. The square root of a negative number has no real solution. 121

The Discriminant Example 4 = ± 2 (2) (2) = 4 and ( 2)( 2) = 4 So BOTH 2 and 2 are solutions 122

The Discriminant What is the relationship between the discriminant of a quadratic and its graph? y = x 2 8x + 10 y = 3x 2 + 8x 4 Discriminant (8) 2 4(1)(10) = 64 40 = 24 ( 6) 2 4(3)( 4) = 36 + 48 = 84 123

The Discriminant What is the relationship between the discriminant of a quadratic and its graph? y = 2x 2 4x + 2 y = x 2 + 6x + 9 Discriminant ( 4) 2 4(2)(2) = 16 16 = 0 (6) 2 4(1)(9) = 36 36 = 0 124

The Discriminant What is the relationship between the discriminant of a quadratic and its graph? y = x 2 + 5x + 9 y = 3x 2 3x + 4 Discriminant (5) 2 4(1)(9) = 25 36 = 11 ( 3) 2 4(3)(4) = 9 48 = 39 125

52 What is value of the discriminant of 2x 2 3x + 5 = 0? Answer 126

53 Find the number of solutions using the discriminant for 2x 2 3x + 5 = 0 A 0 B 1 C 2 Answer 127

54 What is value of the discriminant of x 2 8x + 4 = 0? Answer 128

55 Find the number of solutions using the discriminant for x 2 8x + 4 = 0 A 0 B 1 C 2 Answer 129

Solve Quadratic Equations by Using the Quadratic Formula Return to Table of Contents 130

Solve Any Quadratic Equation At this point you have learned how to solve quadratic equations by: graphing factoring using square roots and completing the square Many quadratic equations may be solved using these methods; however, some cannot be solved using any of these methods. Today we will be given a tool to solve ANY quadratic equation. It ALWAYS works. 131

The Quadratic Formula The solutions of ax 2 + bx + c = 0, where a 0, are: x = b ± b 2 4ac 2a "x equals the opposite of b, plus or minus the square root of b squared minus 4ac, all divided by 2a." 132

The Quadratic Formula Example 1 2x 2 + 3x 5 = 0 2x 2 + 3x + ( 5) = 0 Identify values of a, b and c x = b ± b 2 4ac 2a Write the Quadratic Formula x = 3 ± 3 2 4(2)( 5) 2(2) Substitute the values of a, b and c continued on next slide 133

The Quadratic Formula x = 3 ± 9 ( 40) 4 Simplify x = 3 ± 49 4 = 3 ± 7 4 Write as two equations x = 3 + 7 x = 3 7 4 or 4 x = 1 or x = 5 2 Solve each equation 134

The Quadratic Formula Example 2 2x = x 2 3 Remember - In order to use the Quadratic Formula, the equation must be in standard form (ax 2 + bx +c = 0). First, rewrite the equation in standard form. 2x = x 2 3 2x 2x Use only addition for standard form 0 = x 2 + (-2x) + ( 3) x 2 + ( 2x) + ( 3) = 0 Flip the equation Now you are ready to use the Quadratic Formula Solution on next slide 135

x 2 + ( 2x) + ( 3) = 0 The Quadratic Formula 1x 2 + ( 2x) + ( 3) = 0 Identify values of a, b and c x = b ± b 2 4ac 2a x = ( 2) ± ( 2) 2 4(1)( 3) 2(1) Write the Quadratic Formula Substitute the values of a, b and c Continued on next slide 136

The Quadratic Formula x = 2 ± 4 ( 12) 2a Simplify x = 2 ± 16 2 = 2 ± 4 2 x = 2 ± 4 2 or x = 2-4 2 Write as two equations x = 3 or x = 1 Solve each equation 137

56 Solve the following equation using the quadratic formula: x 2 5x + 4 = 0 A -5 B -4 C -3 D -2 E -1 F 1 G 2 H 3 I 4 J 5 Answer 138

57 Solve the following equation using the quadratic formula: x 2 = x + 20 A 5 B 4 C 3 D 2 E 1 F 1 G 2 H 3 I 4 J 5 Answer 139

58 Solve the following equation using the quadratic formula: 2x 2 + 12 = 11x A 5 B 4 3 C 2 D 2 E 1 F 1 G 2 3 H 2 I 4 J 5 Answer 140

The Quadratic Formula Example 3 x 2 2x 4 = 0 1x 2 + ( 2x) + ( 4) = 0 Identify values of a, b and c x = -b ± b 2-4ac 2a Write the Quadratic Formula x = ( 2) ± ( 2) 2 4(1)( 4) 2(1) Substitute the values of a, b and c Continued on next slide 141

The Quadratic Formula x = 2 ± 4 ( 16) 2 Simplify x = 2 ± 20 2 x = 2 ± 20 2 or x = 2-20 2 x = 2 ± 2 5 2 or x = 2-2 5 2 Write as two equations x = 1 + 5 or x = 1 5 x 3.24 or x 1.24 Use a calculator to estimate x 142

59 Find the larger solution to x 2 + 6x 1 = 0 Answer 143

60 Find the smaller solution to x 2 + 6x 1 = 0 Answer 144

Application Problems Return to Table of Contents 145

Quadratic Equations and Applications A sampling of applied problems that lend themselves to being solved by quadratic equations: Number Reasoning Distances Geometry: Dimensions Free Falling Objects Height of a Projectile 146

Number Reasoning The product of two consecutive negative integers is 1,122. What are the numbers? Remember that consecutive integers are one unit apart, so the numbers are n and n + 1. Multiplying to get the product: n(n + 1) = 1122 n 2 + n = 1122 n 2 + n 1122 = 0 (n + 34)(n - 33) = 0 n = 34 and n = 33. STANDARD Form FACTOR The solution is either 34 and 33 or 33 and 34, since the direction ask for negative integers 34 and 33 are the correct pair. 147

Application Problems PLEASE KEEP THIS IN MIND When solving applied problems that lead to quadratic equations, you might get a solution that does not satisfy the physical constraints of the problem. For example, if x represents a width and the two solutions of the quadratic equations are 9 and 1, the value 9 is rejected since a width must be a positive number. 148

61 The product of two consecutive even integers is 48. Find the smaller of the two integers. Hint: x(x+2) = 48 Click to reveal hint Answer 149

Application Problems TRY THIS: The product of two consecutive integers is 272. What are the numbers? 150

62 The product of two consecutive even integers is 528. What is the smaller number? Answer 151

More of a challenge... The product of two consecutive odd integers is 1 less than four times their sum. Find the two integers. Let n = 1st number n + 2 = 2nd number n(n + 2) = 4[n + (n + 2)] 1 n 2 + 2n = 4[2n + 2] 1 n 2 + 2n = 8n + 8 1 n 2 + 2n = 8n + 7 n 2 6n - 7 = 0 (n 7)(n + 1) = 0 n = 7 and n = 1 Which one do you use? Or do you use both? 152

More of a challenge... If n = 7 then n + 2 = 9 7 x 9 = 4[7 + (7 + 2)] 1 63 = 4(16) 1 63 = 64 1 63 = 63 If n = 1 then n + 2 = 1 + 2 = 1 ( 1) x 1 = 4[ 1 + ( 1 + 2)] 1 1 = 4[ 1 + 1] 1 1 = 4(0) 1 1 = 1 We get two sets of answers. 153

63 The product of a number and a number 3 more than the original is 418. What is the smallest value the original number can be? Answer 154

64 Find three consecutive positive even integers such that the product of the second and third integers is twenty more than ten times the first integer. Enter th e value of the smaller even integer. Answer From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011. 155

65 When 36 is subtracted from the square of a number, the result is five times the number. What is the positive solution? A 9 B 6 Answer C 3 D 4 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011. 156

66 Tamara has two sisters. One of the sisters is 7 years older than Tamara.The other sister is 3 years younger than Tamara. The product of Tamara s sisters ages is 24. How old is Tamara? Answer From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011. 157

Distance Problems Example Two cars left an intersection at the same time, one heading north and one heading west. Some time later, they were exactly 100 miles apart. The car that headed north had gone 20 miles farther than the car headed west. How far had each car traveled? Step 1 - Read the problem carefully. Step 2 - Illustrate or draw your information. 100 x+20 Step 3 - Assign a variable Let x = the distance traveled by the car heading x west Then (x + 20) = the distance traveled by the car heading north Step 4 - Write an equation Does your drawing remind you of the Pythagorean Theorem? a 2 + b 2 = c 2 Continued on next slide 158

Distance Problems Step 5 - Solve a 2 + b 2 = c 2 x 2 + (x+20) 2 = 100 2 100 x x+20 x 2 + x 2 + 40x + 400 = 10,000 2x 2 + 40x 9600 = 0 2(x 2 +20x 4800) = 0 x 2 + 20x 4800 = 0 Square the binomial Standard form Factor Divide each side by 2 Think about your options for solving the rest of this equation. Completing the square? Quadratic Formula? Continued on next slide 159

Distance Problems Did you try the quadratic formula? x = 20 ± 400 4(1)( 4800) 2 x = 20 ± 19,600 2 x = 60 or x = -80 Since the distance cannot be negative, discard the negative solution. The distances are 60 miles and 60 + 20 = 80 miles. Step 6 - Check your answers. 160

67 Two cars left an intersection at the same time,one heading north and the other heading east. Some time later they were 200 miles apart. If the car heading east traveled 40 miles farther, how far did the car traveling north go? Answer 161

Geometry Applications Area Problem The length of a rectangle is 6 inches more than its width. The area of the rectangle is 91 square inches. Find the dimensions of the rectangle. Step 1 - Draw the picture of the rectangle. Let the width = x and the length = x + 6 Step 2 - Write the equation using the formula Area = length x width x + 6 x 162

Geometry Applications Step 3 - Solve the equation x( x + 6) = 91 x 2 + 6x = 91 x 2 + 6x 91 = 0 (x 7)(x + 13) = 0 x = 7 or x = 13 Since a length cannot be negative... The width is 7 and the length is 13. 163

68 The width of a rectangular swimming pool is 10 feet less than its length. The surface area of the pool is 600 square feet. What is the pool's width? Hint: (L)(L 10) = 600. Click to reveal hint Answer 164

69 A square's length is increased by 4 units and its width is increased by 6 units. The result of this transformation is a rectangle with an area that 195 square units. Find the area of the original square. Answer 165

70 The rectangular picture frame below is the same width all the way around. The photo it surrounds measures 17" by 11". The area of the frame and photo combined is 315 sq. in. What is the length of the outer frame? length x Answer x 166

71 The area of the rectangular playground enclosure at South School is 500 square meters. The length of the playground is 5 meters longer than the width. Find the dimensions of the playground, in meters. [Only an algebraic solution will be accepted.] Answer From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011. 167

72 Jack is building a rectangular dog pen that he wishes to enclose. The width of the pen is 2 yards less than the length. If the area of the dog pen is 15 square yards, how many yards of fencing would he need to completely enclose the pen? Answer 168

Free Falling Objects Problems 169

73 A person walking across a bridge accidentally drops an orange in the river below from a height of 40 ft. The function h = 16t 2 + 40 gives the orange's approximate height h above the water, in feet, after t seconds. In how many t seconds will the orange hit the water? (Round to the nearest tenth.) Hint: when it hits the water it is at 0. Answer 170

74 Greg is in a car at the top of a roller-coaster ride. The distance, d, of the car from the ground as the car descends is determined by the equation d = 144 16t 2 where t is the number of seconds it takes the car to travel down to each point on the ride. How many seconds will it take Greg to reach the ground? Answer 171

75 The height of a golf ball hit into the air is modeled by the equation h = 16t 2 + 48t, where h represents the height, in feet, and t represents the number of seconds that have passed since the ball was hit. What is the height of the ball after 2 seconds? A 16 ft B 32 ft C 64 ft D 80 ft Answer From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011. 172

Height of Projectiles Problems 173

76 A skyrocket is shot into the air. It's altitude in feet, h, after t seconds is given by the function h = 16t 2 + 128t. What is the rocket's maximum altitude? Answer 174

77 A rocket is launched from the ground and follows a parabolic path represented by the equation y = x + 2 10x. At the same time, a flare is launched from a height of 10 feet and follows a straight path represented by the equation y = x + 10. Using the accompanying set of axes, graph the equations that represent the paths of the rocket and the flare, and find the coordinates of the point or points where the paths intersect. Answer From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011. 175