Slide 48 (Answer) / 222 10 What is the vertex for y = x 2 + 2x - 3 (Step 2)? A (-1, -4) Answer A B (1, -4) C (-1, -6) [This object is a pull tab] D (1, -6) Slide 49 / 222 11 What is the y-intercept for y = x 2 + 2x - 3 (Step 3)? A (0 , -3) B (0 , 3) Slide 49 (Answer) / 222 11 What is the y-intercept for y = x 2 + 2x - 3 (Step 3)? A (0 , -3) B (0 , 3) Answer A [This object is a pull tab]
Slide 50 / 222 Graph Practice: Graph Slide 50 (Answer) / 222 Graph Practice: Graph Answer [This object is a pull tab] Slide 51 / 222 Graph Practice: Graph
Slide 51 (Answer) / 222 Graph Practice: Graph Answer [This object is a pull tab] Slide 52 / 222 Graph Practice: Graph Slide 52 (Answer) / 222 Graph Practice: Graph Answer [This object is a pull tab]
Slide 53 / 222 Solve Quadratic Equations by Graphing Return to Table of Contents Slide 54 / 222 Solve by Graphing When asked to solve a quadratic equation, there are several ways to do so. One way to solve a quadratic equation in standard form is to find the zeros of the related function by graphing. A zero is the point at which the parabola intersects the x-axis. A quadratic function may have one, two or no zeros. Slide 55 / 222 Solve by Graphing How many zeros do the parabolas have? What are the values of the zeros? No zeroes 2 zeroes; 1 zero; click click click x = -1 and x=3 x=1
Slide 56 / 222 Vocabulary Every quadratic function has a related quadratic equation. A quadratic equation is used to find the zeroes of a quadratic function. When a function intersects the x-axis its y-value is zero. When writing a quadratic function as its related quadratic equation, you replace y with 0. So y = 0. y = ax 2 + bx + c Quadratic Function 0 = ax 2 + bx + c ax 2 + bx + c = 0 Quadratic Equation Slide 57 / 222 Solve by Graphing One way to solve a quadratic equation in standard form is to find the zeros or x-intercepts of the related function. Solve a quadratic equation by graphing: Step 1 - Write the related function. Step 2 - Graph the related function. Step 3 - Find the zeros (or x-intercepts) of the related function. Slide 58 / 222 Solve by Graphing Step 1 - Write the Related Function 2x 2 - 18 = 0 2x 2 - 18 = y y = 2x 2 + 0x - 18
Slide 59 / 222 Solve by Graphing Step 2 - Graph the Function y = 2x 2 + 0x – 18 Use the same five-step process for graphing The axis of symmetry is x = 0. The vertex is (0, -18). The y-intercept is (0, -18). Since the vertex is the y-intercept, locate two other points by substituting values for x. We'll use (2,-10) and (3,0) Graph these points and use reflection across the axis of symmetry. Connect all points with a smooth curve. Slide 59 (Answer) / 222 Solve by Graphing The vertex is found by Step 2 - Graph the Function substituting the Vertex x-coordinate of the Axis of Symmetry into the y = 2x 2 + 0x – 18 equation and solving for y. Use the same five-step process for graphing Two Points Find two more points on the same Hint side of the Axis of Symmetry as the point containing the The axis of symmetry is x = 0. y-intercept. The vertex is (0, -18). The y-intercept is (0, -18). Since the vertex is the y-intercept, locate two The point where the line t passes through the y-axis. p other points by substituting values for x. We'll e c This occurs when the r e x-value is 0. t use (2,-10) and (3,0) n I Y [This object is a pull tab] Graph these points and use reflection across the axis of symmetry. Connect all points with a smooth curve. Slide 60 / 222 Solve by Graphing Step 2 - Graph the Function y = 2x 2 + 0x – 18 x = 0 # # (3,0) # # (2,-10) # (0,-18)
Slide 61 / 222 Solve by Graphing Step 3 - Find the zeros y = 2x 2 + 0x – 18 x = 0 # # (3,0) The zeros appear to be 3 and -3. # # (2,-10) # (0,-18) Slide 62 / 222 Solve by Graphing Step 3 - Find the zeros y = 2x 2 + 0x – 18 Substitute 3 and -3 for x in the quadratic equation. Check 2x 2 – 18 = 0 2(3) 2 – 18 = 0 2(-3) 2 – 18 = 0 2(9) - 18 = 0 2(9) - 18 = 0 18 - 18 = 0 18 - 18 = 0 # 0 = 0 # 0 = 0 The zeros are 3 and -3. Slide 63 / 222 12 Solve the equation by graphing the related function and identifying the zeros. -12x + 18 = -2x 2 Step 1: Which of these is the related function? y = -2x 2 - 12x + 18 A B y = 2x 2 - 12x - 18 y = -2x 2 + 12x - 18 C
Slide 63 (Answer) / 222 12 Solve the equation by graphing the related function and identifying the zeros. -12x + 18 = -2x 2 Step 1: Which of these is the related function? Answer C y = -2x 2 - 12x + 18 A B y = 2x 2 - 12x - 18 C y = -2x 2 + 12x - 18 [This object is a pull tab] Slide 64 / 222 13 What is the axis of symmetry? y = -2x 2 + 12x - 18 Formula: -b A x = -3 2a x = 3 B C x = 4 D x = -5 Slide 64 (Answer) / 222 13 What is the axis of symmetry? y = -2x 2 + 12x - 18 Formula: -b x = -3 A 2a x = 3 B Answer B x = 4 C D x = -5 [This object is a pull tab]
Slide 65 / 222 y = -2x 2 + 12x - 18 14 What is the vertex? (3,0) A (-3,0) B (4,0) C D (-5,0) Slide 65 (Answer) / 222 y = -2x 2 + 12x - 18 14 What is the vertex? A (3,0) Answer A B (-3,0) C (4,0) (-5,0) D [This object is a pull tab] Slide 66 / 222 y = -2x 2 + 12x - 18 15 What is the y-intercept? (0,0) A (0, 18) B C (0, -18) (0, 12) D
Slide 66 (Answer) / 222 y = -2x 2 + 12x - 18 15 What is the y-intercept? (0,0) Answer A C (0, 18) B (0, -18) C [This object is a pull tab] D (0, 12) Slide 67 / 222 16 If two other points are (5, -8) and (4, -2), what does the graph of y = -2x 2 + 12x - 18 look like? B A C D Slide 67 (Answer) / 222 16 If two other points are (5, -8) and (4, -2), what does the graph of y = -2x 2 + 12x - 18 look like? B A Answer C [This object is a pull tab] C D
Slide 68 / 222 y = -2x 2 + 12x - 18 17 Find the zero(s) -18 A 4 B C 3 D -8 Slide 68 (Answer) / 222 17 y = -2x 2 + 12x - 18 Find the zero(s) A -18 Answer B 4 C 3 C D -8 [This object is a pull tab] Slide 69 / 222 Solve Quadratic Equations by Factoring Return to Table of Contents
Slide 69 (Answer) / 222 Please Note: This is not meant to be the students first lesson on Teacher Notes factoring. The emphasis here is factoring followed by using the Zero Solve Quadratic Product Property to find the zeros of the quadratic. There are various ways to teach factoring. It is advantageous to show students Equations by multiple methods as they can then choose which works best for them. [This object is a teacher notes pull tab] Factoring Return to Table of Contents Slide 70 / 222 Solve by Factoring In addition to graphing, there are additional ways to find the zeros or x-intercepts of a quadratic. This section will explore solving quadratics using the method of factoring. A complete review of factoring can be found in the Fundamental Skills of Algebra (Supplemental Review) Unit. Fundamental Skills of Algebra (Supplemental Review) Click for Link Slide 71 / 222 Solve by Factoring Review of factoring - Factoring is simply rewriting an expression in an equivalent form which uses multiplication. To factor a quadratic, ensure that you have the quadratic in standard form: ax 2 +bx+c=0 Tips for factoring quadratics: · Check for a GCF (Greatest Common Factor). · Check to see if the quadratic is a Difference of Squares or other special binomial product.
Slide 72 / 222 Solve by Factoring Examples: Quadratics with a GCF: 3x 2 + 6x in factored form is 3x(x + 2) Quadratics using Difference of Squares: x 2 - 64 in factored form is (x + 8)(x - 8) Additional Quadratic Trinomials: x 2 - 12x +27 in factored form is (x - 9)(x - 3) 2x 2 - x - 6 in factored form is (2x + 3)(x - 2) Slide 73 / 222 Solve by Factoring Practice: 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 of 18 whose sum is 9. (In other words, find 2 numbers that multiply to 18 but also add to 9.) Factors of 18 Sum Slide 73 (Answer) / 222 Solve by Factoring Practice: To factor a quadratic trinomial of the form x 2 + bx + c, find two factors of c whose sum is b. Sum Example - Factors of 18 To factor x 2 + 9x + 18, look for factors of 18 whose sum is 9. 1 and 18 19 (In other words, find 2 numbers that multiply to 18 but also add to 9.) Answer 2 and 9 11 3 and 6 9 Sum Factors of 18 x 2 + 9x + 18 = (x + 3)(x + 6) [This object is a pull tab]
Slide 74 / 222 Solve by Factoring Practice: Factor x 2 + 4x - 12, look for factors of -12 whose sum is 4. (in other words, find 2 numbers that multiply to -12 but also add to 4.) Factors of -12 Sum Slide 74 (Answer) / 222 Solve by Factoring Practice: Factor x 2 + 4x - 12, look for factors of -12 whose sum is 4. Sum (in other words, find 2 numbers that multiply to -12 but also add to 4.) Factors of -12 -1 and 12 11 Answer Factors of -12 Sum -2 and 6 4 -3 and 4 1 1 and -12 -11 2 and -6 -4 3 and -4 -1 x 2 + 4x - 12 = (x - 2)(x + 6) [This object is a pull tab] Slide 75 / 222 Zero Product Property Imagine this: If 2 numbers must be placed in the boxes and you know that when you multiply these you get ZERO, what must be true? = 0 ? ? x
Slide 75 (Answer) / 222 Zero Product Property Imagine this: If 2 numbers must be placed in the boxes and you know that when you multiply these you get ZERO, what must be This slide is meant to encourage Teacher Notes true? discussion among students. Hopefully, most will respond and realize that one or both of the numbers in the boxes MUST be = 0 zero. This will help with their understanding of the Zero Product ? ? Property using binomials. x [This object is a pull tab] Slide 76 / 222 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. If a b = 0 then a = 0 or b = 0 Slide 77 / 222 Solve by Factoring Now... combining the 2 ideas of factoring with the Zero Product Property, we are able to solve for the x-intercepts (zeros) of the quadratic. Example: Solve x 2 + 4x - 12 = 0 1. Factor the trinomial. 2. Using the Zero Product Property, set each factor equal to zero. 3. Solve each simple equation.
Slide 77 (Answer) / 222 Solve by Factoring Now... combining the 2 ideas of factoring with the Zero Product Property, we are able to solve for the x-intercepts (zeros) of the x 2 + 4x - 12 = 0 quadratic. (x + 6) (x - 2) = 0 Answer Example: Solve x 2 + 4x - 12 = 0 x + 6 = 0 or x - 2 = 0 x = -6 x = 2 1. Factor the trinomial. 2. Using the Zero Product Property, set each factor equal to zero. [This object is a pull tab] 3. Solve each simple equation. Slide 78 / 222 Solve by Factoring Remember: The equation Example: Solve x 2 + 36 = 12x has to be written in standard form (ax 2 + bx + c). Slide 78 (Answer) / 222 Solve by Factoring Remember: The equation Example: Solve x 2 + 36 = 12x has to be written in x 2 + 36 = 12x standard form (ax 2 + bx + -12 x -12x c). Answer x 2 - 12x + 36 = 0 (x - 6)(x - 6) = 0 x - 6 = 0 +6 +6 x = 6 [This object is a pull tab]
Slide 79 / 222 18 Solve x = -7 F x = 3 A x = -5 x = 5 B G x = 6 C x = -3 H D x = -2 I x = 7 x = 2 x = 15 E J Slide 79 (Answer) / 222 18 Solve x = 3 A x = -7 F x = -5 G x = 5 B Answer E, F x = -3 H x = 6 C x = -2 x = 7 D I x = 2 x = 15 E J [This object is a pull tab] Slide 80 / 222 19 Solve m = 3 A m = -7 F m = 5 m = -5 G B C m = -3 H m = 6 m = -2 I m = 7 D m = 2 m = 15 E J
Slide 80 (Answer) / 222 19 Solve m = 3 A m = -7 F m = -5 G m = 5 B Answer B m = -3 H m = 6 C m = -2 m = 7 D I m = 15 E m = 2 J [This object is a pull tab] Slide 81 / 222 20 Solve h = 3 A h = -12 F h = -4 G h = 4 B h = -3 H h = 6 C h = -2 h = 8 D I h = 12 E h = 2 J Slide 81 (Answer) / 222 20 Solve h = -12 F h = 3 A Answer h = -4 h = 4 C, G B G h = 6 C h = -3 H h = 8 D h = -2 I h = 2 J h = 12 E [This object is a pull tab]
Slide 82 / 222 21 Solve d = -7 d = 3 A F d = 5 B d = -5 G d = 6 C d = -3 H D d = -2 I d = 7 d = 0 d = 37 E J Slide 82 (Answer) / 222 21 Solve d = 3 A d = -7 F Answer d = -5 G d = 5 B E, J d = -3 H d = 6 C d = -2 d = 7 D I d = 37 E d = 0 J [This object is a pull tab] Slide 83 / 222 Berry Method to Factor Example: Solve When a does not equal 1, check first for a GCF, then use the Berry Method. Berry Method to factor Step 1: Calculate ac . Step 2: Find a pair of numbers m and n , whose product is ac , and whose sum is b . Step 3: Create the product . Step 4: From each binomial in step 3, factor out and discard any common factor. The result is the factored form.
Slide 84 / 222 Berry Method to Factor Solve Use the Berry Method. a = 8, b = 2, c = -3 Step 1 -4 and 6 are factors of -24 that add to +2 Step 2 Step 3 Step 4 Discard common factors Slide 85 / 222 Berry Method to Factor Solve Use the Zero Product Rule to solve. Slide 86 / 222 Berry Method to Factor Solve Use the Berry Method. a = 4, b = -15, c = -25
Slide 87 / 222 Berry Method to Factor Solve Use the Zero Product Rule to solve. Slide 88 / 222 Berry Method to Factor Solve Slide 88 (Answer) / 222 Berry Method to Factor Solve Answer or [This object is a pull tab]
Slide 89 / 222 Application of the Zero Product Property In addition to finding the x-intercepts of quadratic equations, the Zero Product Property can also be used to solve real world application problems. Return to Table of Contents Slide 90 / 222 Application Example: A garden has a length of (x+7) feet and a width of (x +3) feet. The total area of the garden is 396 sq. ft. Find the width of the garden. Slide 90 (Answer) / 222 Application Example: A garden has a length of (x+7) feet and a width of (x (x+7)(x+3) = 396 +3) feet. The total area of the garden is 396 sq. ft. Find the x 2 + 10x + 21 = 396 x 2 + 10x - 375 = 0 width of the garden. Answer (x-15)(x+25)=0 x=15 x=-25 x=15 feet width = 15+3 = 18 feet [This object is a pull tab]
Slide 91 / 222 22 The product of two consecutive even integers is 48. Find the smaller of the two integers. Hint: Two consecutive integers can be expressed as x and x + 1. Two consecutive even integers can be expressed as x and x + 2. Slide 91 (Answer) / 222 22 The product of two consecutive even integers is 48. Find the smaller of the two integers. Hint: Two consecutive integers can be expressed as x(x+2)=48 x 2 + 2x = 48 x and x + 1. Two consecutive even integers can be x 2 +2x - 48 = 0 (x - 6)(x + 8) = 0 expressed as x and x + 2. x = 6 x=-8 Answer x = 6, 1st integer x + 2 = 8, 2nd integer OR x = -8, 1st integer x + 2 = -6, 2nd integer [This object is a pull tab] Slide 92 / 222 23 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?
Slide 92 (Answer) / 222 23 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? L(L - 10)=600 L 2 - 10L = 600 Answer L 2 - 10L - 600 = 0 (L - 30)(L + 20) = 0 L= 30 L= -20 Length = 30 feet Width = L- 10 = 20 feet [This object is a pull tab] Slide 93 / 222 24 A science class designed a ball launcher and tested it by shooting a tennis ball straight up from the top of a 15-story building. They determined that the motion of the ball could be described by the function: , where t represents the time the ball is in the air in seconds and h(t) represents the height, in feet, of the ball above the ground at time t. What is the maximum height of the ball? At what time will the ball hit the ground? Find all key features and graph the function. Students type their answers here Problem is from: Click link for exact lesson. Slide 93 (Answer) / 222 24 A science class designed a ball launcher and tested it by shooting a tennis ball straight up from the top of a 15-story building. They determined that the motion of the ball could be described by the function: , where t represents the time the ball is in the air in seconds and h(t) represents the height, in feet, of the ball above the ground at time t. What is the maximum height of the ball? At what time will the ball hit the ground? Find Answer all key features and graph the function. Students type their answers here Problem is from: [This object is a pull tab] Click link for exact lesson.
Slide 94 / 222 25 A ball is thrown upward from the surface of Mars with an initial velocity of 60 ft/sec. What is the ball's maximum height above the surface before it starts falling back to the surface? Graph the function. The equation for "projectile motion" on Mars is: Students type their answers here Slide 94 (Answer) / 222 25 A ball is thrown upward from the surface of Mars with an initial velocity of 60 ft/sec. What is the ball's maximum height above the surface before it starts falling back to the surface? Graph the function. The Maximum Height = 138.5 ft equation for "projectile motion" on Mars is: Students type their answers here Answer [This object is a pull tab] Slide 95 / 222 Solve Quadratic Equations Using Square Roots Return to Table of Contents
Slide 95 (Answer) / 222 Note: In this section it is important Solve Quadratic to keep reminding students that Teacher Notes when they take the square root of both sides of an equation, they Equations Using must consider both the positive and negative answers. Many mistakes when solving these questions result in students only Square Roots using the positive square root. [This object is a pull tab] Return to Table of Contents Slide 96 / 222 Slide 97 / 222
Slide 98 / 222 Solve Using Square Roots What if x 2 has a coefficient other than 1? Example: Solve 4x 2 = 20 using the square roots method. Slide 98 (Answer) / 222 Solve Using Square Roots What if x 2 has a coefficient other than 1? Example: Solve 4x 2 = 20 using the square roots method. 4x 2 = 20 4 4 Answer x 2 = 5 x = ±√5 [This object is a pull tab] Slide 99 / 222 26 When you take the square root of a real number, your answer will always be positive. True False
Slide 99 (Answer) / 222 26 When you take the square root of a real number, your answer will always be positive. True Answer False False [This object is a pull tab] Slide 100 / 222 27 If x 2 = 16, then x = A 4 B 2 C -2 D 26 -4 E Slide 100 (Answer) / 222 27 If x 2 = 16, then x = A 4 A B 2 Answer -2 C E D 26 [This object is a pull E -4 tab]
Slide 101 / 222 28 Solve using the square root method. A 5 -5 E B 20 F 2 G -4 C 4 D -2 H -20 Slide 101 (Answer) / 222 28 Solve using the square root method. A 5 E -5 D Answer B 20 F 2 F C 4 G -4 D -2 H -20 [This object is a pull tab] Slide 102 / 222 29 If y 2 = 4, then y = A 4 B 2 -2 C D 26 E -4
Slide 102 (Answer) / 222 29 If y 2 = 4, then y = A 4 B B 2 Answer C -2 C D 26 [This object is a pull tab] E -4 Slide 103 / 222 Slide 103 (Answer) / 222
Slide 104 / 222 Slide 104 (Answer) / 222 Slide 105 / 222 32 If (3g - 9) 2 + 7= 43, then g = A B C D E
Slide 105 (Answer) / 222 32 If (3g - 9) 2 + 7= 43, then g = A B A Answer C D D [This object is a pull tab] E Slide 106 / 222 Solve Using Square Roots Challenge: Solve (2x - 1)² = 20 using the square root method. Slide 106 (Answer) / 222 Solve Using Square Roots Challenge: Solve (2x - 1)² = 20 using the square root method. Answer [This object is a pull tab]
Slide 107 / 222 33 A physics teacher put a ball at the top of a ramp and let it roll toward the floor. The class determined that the height of the ball could be represented by the equation, ,where the height, h, is measured in feet from the ground and time, t, in seconds. Determine the time it takes the ball to reach the floor. Students type their answers here Problem is from: Click link for exact lesson. Slide 107 (Answer) / 222 33 A physics teacher put a ball at the top of a ramp and let it roll toward the floor. The class determined that the height of the ball could be represented by the equation, ,where the height, h, is measured in feet from the ground and time, t, in seconds. Determine the time it takes the ball to Use the lesson on Engage NY Teacher Notes to show two separate methods reach the floor. Students type their answers here for discussion with the students. Answer: 1/2 second [This object is a pull tab] Problem is from: Click link for exact lesson. Slide 108 / 222 34 A rock is dropped from a 1000 foot tower. The height of the rock as a function of time can be modeled by the equation: . How long does it take for the rock to reach the ground? Students type their answers here Answer
Slide 109 / 222 Solving Quadratic Equations by Completing the Square Return to Table of Contents Slide 110 / 222 Completing the Square In algebra, "Completing the Square" is a technique for changing a quadratic expression from standard form: ax 2 + bx + c to the vertex/graphing form: a(x + h) 2 + k. It can also be used as a method for solving quadratic equations. Slide 111 / 222
Slide 112 / 222 Slide 112 (Answer) / 222 Slide 113 / 222 35 Find ( b / 2 ) 2 if b = 14.
Slide 113 (Answer) / 222 35 Find ( b / 2 ) 2 if b = 14. Answer 49 [This object is a pull tab] Slide 114 / 222 36 Find ( b / 2 ) 2 if b = 10. Slide 114 (Answer) / 222 36 Find ( b / 2 ) 2 if b = 10. Answer 25 [This object is a pull tab]
Slide 115 / 222 37 Find ( b / 2 ) 2 if b = -12. Slide 115 (Answer) / 222 37 Find ( b / 2 ) 2 if b = -12. Answer 36 [This object is a pull tab] Slide 116 / 222 38 Complete the square to form a perfect square trinomial. x 2 + 18x + ?
Slide 116 (Answer) / 222 38 Complete the square to form a perfect square trinomial. x 2 + 18x + ? Answer 81 [This object is a pull tab] Slide 117 / 222 39 Complete the square to form a perfect square trinomial. x 2 - 6x + ? Slide 117 (Answer) / 222 39 Complete the square to form a perfect square trinomial. x 2 - 6x + ? Answer 9 [This object is a pull tab]
Slide 118 / 222 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. Slide 119 / 222 Completing the Square Let's look at an example to solve: x 2 + 14x -15 = 0 Step 1 - Rewrite Equation Step 2 - Find (b/2) 2 Step 3 - Add the result to both sides Step 4 - Factor & Simplify Step 5 - Take Square Root of both sides Step 6 - Write 2 Equations & Solve How can you check your solutions? Slide 119 (Answer) / 222 Completing the Square Let's look at an example to solve: x 2 + 14x -15 = 0 Step 1 - Rewrite Equation STEP 1: x 2 + 14x = 15 STEP 2: (14 ÷ 2) 2 = 49 Step 2 - Find (b/2) 2 STEP 3: x 2 + 14x + 49 = 15 + 49 Step 3 - Add the result to both sides STEP 4: (x + 7) 2 = 64 Answer STEP 5: x + 7 = ±8 Step 4 - Factor & Simplify STEP 6: x + 7 = 8 or x + 7 = -8 Step 5 - Take Square Root of both sides SOLUTION: x = 1 or x = -15 Step 6 - Write 2 Equations & Solve [This object is a pull tab] How can you check your solutions?
Slide 120 / 222 Completing the Square Let's look at an example to solve: x 2 - 2x - 2 = 0 Step 1 - Rewrite Equation Step 2 - Find (b/2) 2 Step 3 - Add the result to both sides Step 4 - Factor & Simplify Step 5 - Take Square Root of both sides Step 6 - Write 2 Equations & Solve How can you check your solutions? Slide 120 (Answer) / 222 Completing the Square Let's look at an example to solve: x 2 - 2x - 2 = 0 Step 1 - Rewrite Equation STEP 1: x 2 - 2x = 2 STEP 2: (-2 ÷ 2) 2 = (-1) 2 = 1 Step 2 - Find (b/2) 2 STEP 3: x 2 - 2x + 1 = 2 + 1 Step 3 - Add the result to both sides STEP 4: (x - 1) 2 = 3 Answer STEP 5: x - 1 = ± √3 Step 4 - Factor & Simplify STEP 6: x - 1 = √3 or x - 1 = -√3 SOLUTION: x = 1 + √3 or x = 1 - √3 Step 5 - Take Square Root of both sides [This object is a pull Step 6 - Write 2 Equations & Solve tab] How can you check your solutions? Slide 121 / 222 40 Solve the following by completing the square : x 2 + 6x = -5 A -5 -2 B C -1 D 5 E 2
Slide 121 (Answer) / 222 40 Solve the following by completing the square : x 2 + 6x = -5 A -5 A Answer -2 B C C -1 [This object is a pull tab] 5 D E 2 Slide 122 / 222 41 Solve the following by completing the square : x 2 - 8x = 20 A -10 B -2 C -1 D 10 E 2 Slide 122 (Answer) / 222 41 Solve the following by completing the square : x 2 - 8x = 20 B A -10 Answer -2 B D C -1 [This object is a pull tab] D 10 E 2
Slide 123 / 222 Slide 123 (Answer) / 222 Slide 124 / 222 Completing the Square Challenge: 3x 2 - 10x = -3 *Note: There is no GCF to factor out like the previous example. Step 1 - Rewrite Equation Step 2 - Find (b/2) 2 Step 3 - Add the result to both sides Step 4 - Factor & Simplify Step 5 - Take Square Root of both sides Step 6 - Write 2 Equations & Solve
Slide 124 (Answer) / 222 Completing the Square Challenge: 3x 2 - 10x = -3 *Note: There is no GCF to factor out like the Step 1 - Rewrite Equation previous example. Step 2 - Find (b/2) 2 Step 3 - Add the result to both sides Step 4 - Factor & Simplify Step 5 - Take Square Root of both sides Step 6 - Write 2 Equations & Solve Slide 125 / 222 Completing the Square Challenge: 4x 2 - 17x + 4 = 0 *Note: There is no GCF to factor out. Step 1 - Rewrite Equation Step 2 - Find (b/2) 2 Step 3 - Add the result to both sides Step 4 - Factor & Simplify Step 5 - Take Square Root of both sides Step 6 - Write 2 Equations & Solve Slide 125 (Answer) / 222 Completing the Square Challenge: 4x 2 - 17x + 4 = 0 *Note: There is no GCF to factor out. Step 1 - Rewrite Equation Step 2 - Find (b/2) 2 Answer Step 3 - Add the result to both sides Step 4 - Factor & Simplify Step 5 - Take Square Root of both sides Step 6 - Write 2 Equations & Solve [This object is a pull tab]
Slide 126 / 222 Completing the Square Challenge: -6x 2 - 25x - 25 = 0 *Note: There is no GCF to factor out. Slide 126 (Answer) / 222 Slide 127 / 222 43 Solve the following by completing the square : A B C D E
Slide 127 (Answer) / 222 43 Solve the following by completing the square : A A Answer B B C [This object is a pull tab] D E Slide 128 / 222 Solve Quadratic Equations by Using the Quadratic Formula Return to Table of Contents Slide 129 / 222 Solving Quadratics 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. Though completing the square works for any quadratic equation, it can be cumbersome to repeatedly use the algorithm. Today we will be given a tool to solve ANY quadratic equation, and it ALWAYS works!
Slide 130 / 222 Completing the Square Now try Completing the Square on the standard form of a quadratic equation. Step 1 - Rewrite Equation Step 2 - Find (b/2) 2 Step 3 - Add the result to both sides Step 4 - Factor & Simplify Step 5 - Take Square Root of both sides Step 6 - Write 2 Equations & Solve Slide 130 (Answer) / 222 Completing the Square Now try Completing the Square on the standard form of a quadratic equation. Step 1 - Rewrite Equation Teacher Notes see next slide for Step 2 - Find (b/2) 2 solution Step 3 - Add the result to both sides Step 4 - Factor & Simplify Step 5 - Take Square Root of both sides [This object is a pull tab] Step 6 - Write 2 Equations & Solve Slide 131 / 222 Completing the Square Step 1 - Rewrite Equation and factor out a Steps 2 and 3 - Find (b/2) 2 , Add the result to both sides, simplify Step 4 - Factor & Simplify Step 5 - Take Square Root of both sides Step 6 - Solve for x
Slide 132 / 222 Slide 132 (Answer) / 222 Slide 133 / 222 Quadratic Formula Solve 2x 2 + 3x - 5 = 0 Example 1: Once you identify the values of a, b, and c, simply substitute into the quadratic formula and simplify as much as possible. a = 2 b = 3 c = -5 How can you check your answers?
Slide 133 (Answer) / 222 Slide 134 / 222 Quadratic Formula Solve: 2x = x 2 - 3 Example 2: To use the Quadratic Formula, the equation must be in standard form (ax 2 + bx +c = 0). Identify a, b, and c, then substitue into the formula and simplify. Don't forget to check your results! Slide 134 (Answer) / 222 Quadratic Formula Example 2: Solve: 2x = x 2 - 3 To use the Quadratic Formula, the equation must be in standard form (ax 2 + bx +c = 0). Answer Identify a, b, and c, then substitue into the formula and simplify. Don't forget to check your results! [This object is a pull tab]
Slide 135 / 222 44 Solve the following equation using the quadratic formula: 1 F -5 A G 2 -4 B 3 -3 H C 4 I D -2 5 J -1 E Slide 135 (Answer) / 222 44 Solve the following equation using the quadratic formula: F Answer I 1 -5 F A [This object is a pull tab] 2 G B -4 3 H -3 C 4 I -2 D 5 J E -1 Slide 136 / 222 45 Solve the following equation using the quadratic formula: 1 F -5 A 2 G -4 B H 3 -3 C 4 -2 I D 5 J -1 E
Slide 136 (Answer) / 222 45 Solve the following equation using the quadratic formula: B Answer 1 F -5 A J G 2 -4 B [This object is a pull tab] 3 -3 H C 4 I D -2 5 J -1 E Slide 137 / 222 46 Solve the following equation using the quadratic formula: F 1.5 -5 A 2 G B -4 3 H -3 C 4 I -2 D 5 J E -1.5 Slide 137 (Answer) / 222 46 Solve the following equation using the quadratic formula: F Answer 1.5 F -5 A I 2 G -4 B [This object is a pull tab] H 3 -3 C 4 -2 I D 5 J -1.5 E
Slide 138 / 222 Quadratic Formula Example 3: Solve using the quadratic formula, and simplify the result. x 2 - 2x - 4 = 0 Slide 138 (Answer) / 222 Quadratic Formula Example 3: Solve using the quadratic formula, and simplify the result. x 2 - 2x - 4 = 0 Answer [This object is a pull tab] Slide 139 / 222 47 Find the larger solution to the equation.
Slide 139 (Answer) / 222 47 Find the larger solution to the equation. Answer [This object is a pull tab] Slide 140 / 222 48 Find the smaller solution to the equation. Slide 140 (Answer) / 222
Slide 141 / 222 Which Method Work in small groups to solve the quadratic equation using the following different methods. Factoring Quadratic Completing the Graphing Formula Square Slide 141 (Answer) / 222 Which Method Work in small groups to solve the quadratic equation using the following different methods. Have students compare and contrast which method is better to Factoring Quadratic Completing the Graphing Teacher Notes use for this problem. Formula Square *Answer: x = -3 or x = .5 [This object is a teacher notes pull tab] Slide 142 / 222 Which Method Work in small groups to solve the quadratic equation using the following different methods. Factoring Quadratic Completing the Graphing Formula Square
Slide 142 (Answer) / 222 Which Method Work in small groups to solve the quadratic equation using the following different methods. Have students compare and Factoring Quadratic Completing the contrast which method is better to Graphing Teacher Notes Formula Square use for this problem. *Answer: x = -3 or x = .5 [This object is a teacher notes pull tab] Slide 143 / 222 The Discriminant Return to Table of Contents Slide 144 / 222 Solutions Recall what it means to have 0, 1, or 2 solutions/zeros/roots 2 real solutions no real solutions 1 real solution
Slide 145 / 222 The Discriminant At times, it is not necessary to solve for the zeros or roots of a quadratic function, but simply to know how many roots exist (zero, one, or two). The quickest way to determine how many solutions a quadratic has, algebraically, is to calculate what's called the discriminant. It may look familiar, as the discriminant is a part of the quadratic formula. Slide 146 / 222 Slide 147 / 222 The Discriminant Other important tips before practice: · 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.
Slide 148 / 222 Slide 148 (Answer) / 222 Slide 149 / 222 The Discriminant CONCLUSION: If b 2 - 4ac > 0 (POSITIVE) the quadratic has two real solutions click to reveal If b 2 - 4ac = 0 (ZERO) the quadratic has one real solution If b 2 - 4ac < 0 (NEGATIVE) the quadratic has no real solutions
Slide 150 / 222 49 What is the value of the discriminant of 2x 2 - 2x + 3 = 0 ? Slide 150 (Answer) / 222 49 What is the value of the discriminant of 2x 2 - 2x + 3 = 0 ? Answer -20 [This object is a pull tab] Slide 151 / 222 50 Use the discriminant to find the number of solutions for 2x 2 - 2x + 3 = 0 A 0 B 1 C 2
Slide 151 (Answer) / 222 50 Use the discriminant to find the number of solutions for 2x 2 - 2x + 3 = 0 A 0 Answer A B 1 C 2 [This object is a pull tab] Slide 152 / 222 51 What is the value of the discriminant of x 2 - 8x + 4 = 0 ? Slide 152 (Answer) / 222 51 What is the value of the discriminant of x 2 - 8x + 4 = 0 ? Answer 48 [This object is a pull tab]
Slide 153 / 222 52 Use the discriminant to find the number of solutions for x 2 - 8x + 4 = 0 A 0 B 1 C 2 Slide 153 (Answer) / 222 52 Use the discriminant to find the number of solutions for x 2 - 8x + 4 = 0 A 0 Answer C B 1 [This object is a pull tab] C 2 Slide 154 / 222 Vertex Form Return to Table of Contents
Slide 155 / 222 Vertex Form So far, we have been using quadratics in standard form. However, sometimes when graphing, it is more useful to write them in Vertex Form. A quadratic equation in vertex form: Slide 156 / 222 Vertex Form A quadratic function written in vertex form: Vertex Form shows the location of the vertex ( h , k ). The a still tells the direction of opening. And the axis of symmetry is x = h . Example: Find the vertex, direction of opening and the axis of symmetry for the graph of: Slide 156 (Answer) / 222 Vertex Form A quadratic function written in vertex form: Vertex: (6 , 4) Answer Vertex Form shows the location of the vertex ( h , k ). Direction of openness is down The a still tells the direction of opening. And the axis of symmetry is x = h . Axis of Symmetry: x = 6 Example: [This object is a pull Find the vertex, direction of opening and the axis of symmetry for the tab] graph of:
Slide 157 / 222 Vertex Form Find the vertex, direction of openness and the axis of symmetry for each. A. B. C. Slide 157 (Answer) / 222 Vertex Form Find the vertex, direction of openness and the axis of symmetry for each. A. A. B. B. C. C. [This object is a pull tab] Slide 158 / 222 Vertex Form Find the vertex, direction of openness and the axis of symmetry for each. D. E.
Slide 158 (Answer) / 222 Vertex Form Find the vertex, direction of openness and the axis of symmetry for each. D. D. E. E. [This object is a pull tab] Slide 159 / 222 53 Find the vertex for the graph of A B C D Slide 159 (Answer) / 222 53 Find the vertex for the graph of A B C D Answer D [This object is a pull tab]
Slide 160 / 222 54 Find the direction of opening for the graph of A up down B left C right D Slide 160 (Answer) / 222 54 Find the direction of opening for the graph of up A down B left C D right Answer B [This object is a pull tab] Slide 161 / 222
Slide 161 (Answer) / 222 Slide 162 / 222 Slide 162 (Answer) / 222
Slide 163 / 222 57 Give the direction of opening for the graph of A up down B left C right D Slide 163 (Answer) / 222 57 Give the direction of opening for the graph of up A down B left C Answer A D right [This object is a pull tab] Slide 164 / 222 58 Give the axis of symmetry for the graph of
Slide 164 (Answer) / 222 58 Give the axis of symmetry for the graph of Answer x = -9 [This object is a pull tab] Slide 165 / 222 Slide 165 (Answer) / 222
Slide 166 / 222 60 Give the direction of openness of A up down B left C right D Slide 166 (Answer) / 222 60 Give the direction of openness of up A down B left C Answer A D right [This object is a pull tab] Slide 167 / 222 61 The axis of symmetry for the graph of is ______ .
Slide 167 (Answer) / 222 61 The axis of symmetry for the graph of is ______ . Answer x = 0 [This object is a pull tab] Slide 168 / 222 Slide 168 (Answer) / 222
Slide 169 / 222 Slide 169 (Answer) / 222 Slide 170 / 222
Slide 170 (Answer) / 222 Slide 171 / 222 Converting from Standard Form to Vertex Form To convert from standard form to vertex form, we need to recall the method for completing the square. Step 1 - Write the equation in the form y = x 2 + bx + ___ + c - ___ Step 2 - Find (b ÷ 2) 2 Step 3 - Write the result from Step 2 in the first blank and in the second blank. Step 4 - Rewrite the first three terms as a perfect square. Slide 172 / 222
Slide 172 (Answer) / 222 Slide 173 / 222 Slide 173 (Answer) / 222
Slide 174 / 222 65 What is the vertex form of: A B C D Slide 174 (Answer) / 222 65 What is the vertex form of: A B C Answer D A [This object is a pull tab] Slide 175 / 222
Slide 175 (Answer) / 222 Slide 176 / 222 Slide 176 (Answer) / 222
Slide 177 / 222 Comparing Functions Compare the following functions based on information from the equations. What do the graphs have in common? How are they different? Sketch both graphs to confirm your conclusions. Slide 177 (Answer) / 222 Comparing Functions Both quadratic equation have their vertices at (3, 4). However the graph of f(x) has less vertical stretch than the graph of g(x), and the graph of f(x) opens down, Compare the following functions based on information from the whereas the graph of g(x) opens up. equations. What do the graphs have in common? How are g(x) they different? Sketch both graphs to confirm your conclusions. f(x) Answer [This object is a pull tab] Slide 178 / 222 Two Functions Write two different quadratic equations whose graphs have vertices at (3.5, -7).
Slide 179 / 222 Standard Form to Vertex Form What if "a" does not equal 1? Step 1 - Write the equation in the form y = ax 2 + bx +__+ c - __ Step 2 - Factor: y = a(x 2 + ( b / a )x +__)+ c - __ Step 3 - Find ( b / a ÷ 2) 2 Step 4 - Put your answer from Step 3 in the first blank and multiply Step 3 by a to fill in the second blank. Step 5 - Write trinomial as perfect square. Slide 180 / 222 Slide 180 (Answer) / 222
Slide 181 / 222 Slide 181 (Answer) / 222 Slide 182 / 222
Slide 182 (Answer) / 222 Slide 183 / 222 Slide 183 (Answer) / 222
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