CONFIDENTIAL 1 Grade 8 Algebra1 Solving Quadratic Equations by Factoring

CONFIDENTIAL 2 Warm Up Solve each equation by graphing the related function. 1) x = 0 2) x 2 = x ) - x 2 + 8x = 15

CONFIDENTIAL 3 You have solved quadratic equations by graphing. Another method used to solve quadratic equations is to factor and use the Zero Product Property. Zero Product Property Notice that when writing a quadratic function as its related quadratic equation, you replace y with 0. So y = 0. y = ax 2 + bx + c 0 = ax 2 + bx + c ax 2 + bx + c = 0

CONFIDENTIAL 4 One way to solve a quadratic equation in standard form is to graph the related function and find the x- values where y = 0. In other words, find the zeros of the related function. Recall that a quadratic function may have two, one, or no zeros. Using the Zero Product Property WORDSNUMBERSALGEBRA If the product of two quantities equals zero, at least one of the quantities equals zero. 3 (0) = 0 0(4) = 0 If ab = 0, then a = 0 or b = 0.

CONFIDENTIAL 5 Solving Quadratic Equations by Graphing Use the Zero Product Property to solve each equation. Check your answer. A) (x - 3)(x + 7) = 0 x - 3 = 0 or x + 7 = 0 x= 3 or x = -7 Use the Zero Product Property. Solve each equation. Check (x - 3)(x + 7) = 0 (3 - 3)(3 + 7) 0 (0)(10) 0 0 (-7 - 3)(x + 7) = 0 (-7 - 3)(-7 + 7) 0 (10)(0) 0 0 Substitute each solution for x into the original equation. The solutions are 3 and -7.

CONFIDENTIAL 6 B) (x)(x - 5) = 0 x = 0 or x - 5 = 0 x= 5 Use the Zero Product Property. Solve each equation. Check (x)(x - 5) = 0 (0)(0 - 5) 0 (0)(-5) 0 0 Substitute each solution for x into the original equation. The solutions are 0 and 5. (x)(x - 5) = 0 (5)(5 - 5) 0 (5)(0) 0 0

CONFIDENTIAL 7 Now you try! Use the Zero Product Property to solve each equation. Check your answer. 1a. (x)(x + 4) = 0 1b. (x + 4)(x - 3) = 0

CONFIDENTIAL 8 If a quadratic equation is written in standard form, a x 2 + bx + c = 0, then to solve the equation, you may need to factor before using the Zero Product Property.

CONFIDENTIAL 9 Solving Quadratic Equations by Factoring Solve each quadratic equation by factoring. A) x 2 + 7x + 10 = 0 (x + 5) (x + 2) = 0 x + 5 = 0 or x + 2 = 0 Use the Zero Product Property. Solve each equation. Check x 2 + 7x + 10 = 0 (-5) 2 + 7(-5) Substitute each solution for x into the original equation. The solutions are -5 and -2. x = -5 or x = -2 Factor the trinomial. x 2 + 7x + 10 = 0 (-2) 2 + 7(-2)

CONFIDENTIAL 10 B) x 2 + 2x = 8 -8 Use the Zero Product Property. Solve each equation. The solutions are -4 and 2. x = -4 or x = 2 Factor the trinomial. x 2 + 2x = 8 x 2 + 2x – 8 = 0 The equation must be written in standard form. So subtract 8 from both sides. (x + 4) (x - 2) = 0 x + 4 = 0 or x - 2 = 0

CONFIDENTIAL 11 Check: Graph the related quadratic function. The zeros of the related function should be the same as the solutions from factoring. The graph of y = x 2 + 2x - 8 shows two zeros appear to be -4 and 2, the same as the solutions from factoring.

CONFIDENTIAL 12 C) x 2 + 2x + 1 = 0 Use the Zero Product Property. Solve each equation. Both factors result in the same solution, so there is one solution, -1. x = -1 or x = -1 Factor the trinomial. (x + 1) (x + 1) = 0 x + 1 = 0 or x + 1 = 0

CONFIDENTIAL 13 Check: Graph the related quadratic function. The zeros of the related function should be the same as the solutions from factoring. The graph of y = x 2 + 2x + 1 shows that one zero appears to be -1, the same as the solution from factoring.

CONFIDENTIAL 14 D) -2x 2 = x -2 (x - 3) (x - 3) = 0 -2 ≠ 0 or x - 3 = 0 Use the Zero Product Property. Solve each equation. Check The only solution is 3. x = 3 Factor the trinomial. -2x 2 = x -2(3) (3) Write the equation in standard form. -2x x – 18 = 0 -2( x 2 - 6x + 9) = 0 Factor out the GCF, -2. Substitute 3 into the original equation.

CONFIDENTIAL 15 Now you try! Solve each quadratic equation by factoring. Check your answer. 2a. x 2 - 6x + 9 = 0 2b. x 2 + 4x = 5

CONFIDENTIAL 16 Sports Application The height of a diver above the water during a dive can be modeled by h = -16t 2 + 8t + 48, where h is height in feet and t is time in seconds. Find the time it takes for the diver to reach the water. h = -16t 2 + 8t = -16t 2 + 8t + 48 Use the Zero Product Property. Solve each equation. Factor the trinomial. The diver reaches the water when h = 0. Factor out the GCF, = -8(2t 2 - t - 6) 0 = -8(2t + 3) (t -2) -8 ≠ 0, 2t + 3 = 0 or t - 2 = 0 2t = -3 or t = 2 t = -3 2 Since time cannot be negative, (-3/2 ) does not make sense in this situation.

CONFIDENTIAL 17 It takes the diver 2 seconds to reach the water. Check 0 = -16 t 2 + 8t (2) 2 + 8(2) Substitute 3 into the original equation.

CONFIDENTIAL 18 Now you try! 3.) The equation for the height above the water for another diver can be modeled by h = -16t 2 + 8t Find the time it takes this diver to reach the water.

CONFIDENTIAL 19 BREAK

CONFIDENTIAL 21 Assessment 1) (x + 2) (x - 8) = 0 Use the Zero Product Property to solve each equation. Check your answer. 2) (x - 6) (x - 5) = 0 3) (x + 7) (x + 9) = 0 4) (x) (x - 1) = 0

CONFIDENTIAL 22 Solve each quadratic equation by factoring. Check your answer 6) 3x 2 - 4x + 1 = 0 5) 30x = -9x ) x 2 - 8x - 9 = 0 7) x 2 + 4x - 12 = 0

CONFIDENTIAL 23 9) A group of friends tries to keep a beanbag from touching the ground without using their hands. Once the beanbag has been kicked, its height can be modeled by h = -16t t + 2, where h is the height in feet above the ground and t is the time in seconds. Find the time it takes the beanbag to reach the ground.

CONFIDENTIAL 24 You have solved quadratic equations by graphing. Another method used to solve quadratic equations is to factor and use the Zero Product Property. Zero Product Property Notice that when writing a quadratic function as its related quadratic equation, you replace y with 0. So y = 0. y = ax 2 + bx + c 0 = ax 2 + bx + c ax 2 + bx + c = 0 Let’s review

CONFIDENTIAL 25 One way to solve a quadratic equation in standard form is to graph the related function and find the x- values where y = 0. In other words, find the zeros of the related function. Recall that a quadratic function may have two, one, or no zeros. Using the Zero Product Property WORDSNUMBERSALGEBRA If the product of two quantities equals zero, at least one of the quantities equals zero. 3 (0) = 0 0(4) = 0 If ab = 0, then a = 0 or b = 0.

CONFIDENTIAL 26 Solving Quadratic Equations by Graphing Use the Zero Product Property to solve each equation. Check your answer. A) (x - 3)(x + 7) = 0 x - 3 = 0 or x + 7 = 0 x= 3 or x = -7 Use the Zero Product Property. Solve each equation. Check (x - 3)(x + 7) = 0 (3 - 3)(3 + 7) 0 (0)(10) 0 0 (-7 - 3)(x + 7) = 0 (-7 - 3)(-7 + 7) 0 (10)(0) 0 0 Substitute each solution for x into the original equation. The solutions are 3 and -7.

CONFIDENTIAL 27 Solving Quadratic Equations by Factoring Solve each quadratic equation by factoring. A) x 2 + 7x + 10 = 0 (x + 5) (x + 2) = 0 x + 5 = 0 or x + 2 = 0 Use the Zero Product Property. Solve each equation. Check x 2 + 7x + 10 = 0 (-5) 2 + 7(-5) Substitute each solution for x into the original equation. The solutions are -5 and -2. x = -5 or x = -2 Factor the trinomial. x 2 + 7x + 10 = 0 (-2) 2 + 7(-2)

CONFIDENTIAL 28 B) x 2 + 2x = 8 -8 Use the Zero Product Property. Solve each equation. The solutions are -4 and 2. x = -4 or x = 2 Factor the trinomial. x 2 + 2x = 8 x 2 + 2x – 8 = 0 The equation must be written in standard form. So subtract 8 from both sides. (x + 4) (x - 2) = 0 x + 4 = 0 or x - 2 = 0

CONFIDENTIAL 29 Check: Graph the related quadratic function. The zeros of the related function should be the same as the solutions from factoring. The graph of y = x 2 + 2x - 8 shows two zeros appear to be -4 and 2, the same as the solutions from factoring.

CONFIDENTIAL 30 Sports Application The height of a diver above the water during a dive can be modeled by h = -16t 2 + 8t + 48, where h is height in feet and t is time in seconds. Find the time it takes for the diver to reach the water. h = -16t 2 + 8t = -16t 2 + 8t + 48 Use the Zero Product Property. Solve each equation. Factor the trinomial. The diver reaches the water when h = 0. Factor out the GCF, = -8(2t 2 - t - 6) 0 = -8(2t + 3) (t -2) -8 ≠ 0, 2t + 3 = 0 or t - 2 = 0 2t = -3 or t = 2 t = -3 2 Since time cannot be negative, (-3/2 ) does not make sense in this situation.

CONFIDENTIAL 31 It takes the diver 2 seconds to reach the water. Check 0 = -16 t 2 + 8t (2) 2 + 8(2) Substitute 3 into the original equation.

CONFIDENTIAL 32 You did a great job today!