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9-5 Solving Quadratic Equations by Factoring Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview

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9-5 Solving Quadratic Equations by Factoring Warm Up Find each product. 1. (x + 2)(x + 7)2. (x – 11)(x + 5) 3. (x – 10) 2 Factor each polynomial. 4. x 2 + 12x + 355. x 2 + 2x – 63 6. x 2 – 10x + 167. 2x 2 – 16x + 32 x 2 + 9x + 14 x 2 – 6x – 55 x 2 – 20x + 100 (x + 5)(x + 7)(x – 7)(x + 9) (x – 2)(x – 8)2(x – 4) 2

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9-5 Solving Quadratic Equations by Factoring California Standards 14.0 Students solve a quadratic equation by factoring or completing the square. 23.0 Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.

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9-5 Solving Quadratic Equations by Factoring You have solved quadratic equations by graphing. Another method used to solve quadratic equations is to factor and use the Zero Product Property.

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9-5 Solving Quadratic Equations by Factoring Additional Example 1A: Use the Zero Product Property Use the Zero Product Property to solve the equation. Check your answer. (x – 7)(x + 2) = 0 x – 7 = 0 or x + 2 = 0 x = 7 or x = –2 Use the Zero Product Property. Solve each equation.

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9-5 Solving Quadratic Equations by Factoring Additional Example 1A Continued Use the Zero Product Property to solve the equation. Check your answer. Substitute each solution for x in the original equation. Check (x – 7)(x + 2) = 0 (7 – 7)(7 + 2) 0 (0)(9) 0 0 Check (x – 7)(x + 2) = 0 (–2 – 7)(–2 + 2) 0 (–9)(0) 0 0

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9-5 Solving Quadratic Equations by Factoring Additional Example 1B: Use the Zero Product Property Use the Zero Product Property to solve each equation. Check your answer. (x – 2)(x) = 0 x = 0 or x – 2 = 0 x = 2 Use the Zero Product Property. Solve the second equation. Substitute each solution for x in the original equation. Check (x – 2)(x) = 0 (0 – 2)(0) 0 (–2)(0) 0 0 (x – 2)(x) = 0 (2 – 2)(2) 0 (0)(2) 0 0 0 (x)(x – 2) = 0

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9-5 Solving Quadratic Equations by Factoring Use the Zero Product Property to solve each equation. Check your answer. Check It Out! Example 1a (x)(x + 4) = 0 x = 0 or x + 4 = 0 x = –4 Use the Zero Product Property. Solve the second equation. Substitute each solution for x in the original equation. Check (x)(x + 4) = 0 (0)(0 + 4) 0 (0)(4) 0 0 0 (x)(x +4) = 0 (–4)(–4 + 4) 0 (–4)(0) 0 0

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9-5 Solving Quadratic Equations by Factoring Check It Out! Example 1b Use the Zero Product Property to solve the equation. Check your answer. (x + 4)(x – 3) = 0 x + 4 = 0 or x – 3 = 0 x = –4 or x = 3 Use the Zero Product Property. Solve each equation.

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9-5 Solving Quadratic Equations by Factoring Check It Out! Example 1b Continued Use the Zero Product Property to solve the equation. Check your answer. (x + 4)(x – 3) = 0 Substitute each solution for x in the original equation. Check (x + 4)(x – 3 ) = 0 (–4 + 4)(–4 –3) 0 (0)(–7) 0 0 Check (x + 4)(x – 3 ) = 0 (3 + 4)(3 – 3) 0 (7)(0) 0 0

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9-5 Solving Quadratic Equations by Factoring You may need to factor before using the Zero Product Property. You can check your answers by substituting into the original equation or by graphing. If the factored form of the equation has two different factors, the graph of the related function will cross the x-axis in two places. If the factored form has two identical factors, the graph will cross the x-axis in one place.

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9-5 Solving Quadratic Equations by Factoring To review factoring techniques, see Lessons 8-3 through 8-5. Helpful Hint

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9-5 Solving Quadratic Equations by Factoring Additional Example 2A: Solving Quadratic Equations by Factoring Solve the quadratic equation by factoring. Check your answer. x 2 – 6x + 8 = 0 (x – 4)(x – 2) = 0 x – 4 = 0 or x – 2 = 0 x = 4 or x = 2 Factor the trinomial. Use the Zero Product Property. Solve each equation. x 2 – 6x + 8 = 0 (4) 2 – 6(4) + 8 0 16 – 24 + 8 0 0 0 Check x 2 – 6x + 8 = 0 (2) 2 – 6(2) + 8 0 4 – 12 + 8 0 0 0 Check

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9-5 Solving Quadratic Equations by Factoring Additional Example 2B: Solving Quadratic Equations by Factoring Solve the quadratic equation by factoring. Check your answer. x 2 + 4x = 21 –21 x 2 + 4x – 21 = 0 (x + 7)(x –3) = 0 x + 7 = 0 or x – 3 = 0 x = –7 or x = 3 The equation must be written in standard form. So subtract 21 from both sides. Factor the trinomial. Use the Zero Product Property. Solve each equation.

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9-5 Solving Quadratic Equations by Factoring Additional Example 2B Continued Check Graph the related quadratic function. Because there are two solutions found by factoring, the graph should cross the x-axis in two places. The graph of y = x 2 + 4x – 21 intersects the x-axis at x = –7 and x = 3, the same as the solutions from factoring. ●●

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9-5 Solving Quadratic Equations by Factoring Additional Example 2C: Solving Quadratic Equations by Factoring Solve the quadratic equation by factoring. Check your answer. x 2 – 12x + 36 = 0 (x – 6)(x – 6) = 0 x – 6 = 0 or x – 6 = 0 x = 6 or x = 6 Both factors result in the same solution, so there is one solution, 6. Factor the trinomial. Use the Zero Product Property. Solve each equation.

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9-5 Solving Quadratic Equations by Factoring Additional Example 2C Continued Check Graph the related quadratic function. The graph of y = x 2 – 12x + 36 shows one zero at 6, the same as the solution from factoring. ●

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9-5 Solving Quadratic Equations by Factoring Additional Example 2D: Solving Quadratic Equations by Factoring Solve the quadratic equation by factoring. Check your answer. –2x 2 = 20x + 50 The equation must be written in standard form. So add 2x 2 to both sides. Factor out the GCF 2. +2x 2 0 = 2x 2 + 20x + 50 –2x 2 = 20x + 50 2x 2 + 20x + 50 = 0 2(x 2 + 10x + 25) = 0 Factor the trinomial. 2(x + 5)(x + 5) = 0 2 ≠ 0 or x + 5 = 0 x = –5 Use the Zero Product Property. Solve the equation.

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9-5 Solving Quadratic Equations by Factoring Additional Example 2D Continued Solve the quadratic equation by factoring. Check your answer. –2x 2 = 20x + 50 Check –2x 2 = 20x + 50 –2(–5) 2 20(–5) + 50 –50 –100 + 50 –50 Substitute –5 into the original equation.

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9-5 Solving Quadratic Equations by Factoring Check It Out! Example 2a Solve the quadratic equation by factoring. Check your answer. x 2 – 6x + 9 = 0 (x – 3)(x – 3) = 0 x – 3 = 0 or x – 3 = 0 x = 3 or x = 3 The only solution is 3. Factor the trinomial. Use the Zero Product Property. Solve each equation. x 2 – 6x + 9 = 0 (3) 2 – 6(3) + 9 0 9 – 18 + 9 0 0 0 Check Substitute 3 into the original equation.

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9-5 Solving Quadratic Equations by Factoring Check It Out! Example 2b Solve the quadratic equation by factoring. Check your answer. x 2 + 4x = 5 –5 x 2 + 4x – 5 = 0 Write the equation in standard form. Add –5 to both sides. Factor the trinomial. Use the Zero Product Property. Solve each equation. (x – 1)(x + 5) = 0 x – 1 = 0 or x + 5 = 0 x = 1 or x = –5

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9-5 Solving Quadratic Equations by Factoring Check It Out! Example 2b Continued Check Graph the related quadratic function. Because there are two solutions found by factoring, the graph should cross the x-axis in two places. The graph of y = x 2 + 4x – 5 intersects the x-axis at x = 1 and x = –5, the same as the solutions from factoring. ● ●

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9-5 Solving Quadratic Equations by Factoring Check It Out! Example 2c Solve the quadratic equation by factoring. Check your answer. 30x = –9x 2 – 25 –9x 2 – 30x – 25 = 0 –1(3x + 5)(3x + 5) = 0 –1(9x 2 + 30x + 25) = 0 –1 ≠ 0 or 3x + 5 = 0 Write the equation in standard form. Factor the trinomial. Use the Zero Product Property. – 1 cannot equal 0. Solve the remaining equation. Factor out the GCF, –1.

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9-5 Solving Quadratic Equations by Factoring Check It Out! Example 2c Continued Check Graph the related quadratic function. Because there is one solution found by factoring, the graph should cross the x-axis in one place. The graph of y = –9x 2 – 30x – 25 intersects the x-axis at x =, the same as the solution from factoring. ●

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9-5 Solving Quadratic Equations by Factoring Check It Out! Example 2d Solve the quadratic equation by factoring. Check your answer. 3x 2 – 4x + 1 = 0 (3x – 1)(x – 1) = 0 Factor the trinomial. Use the Zero Product Property. Solve each equation. 3x – 1 = 0 or x – 1 = 0 or x = 1

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9-5 Solving Quadratic Equations by Factoring Check It Out! Example 2d Continued 3x 2 – 4x + 1 = 0 3(1) 2 – 4(1) + 1 0 3 – 4 + 1 0 0 0 Check 3x 2 – 4x + 1 = 0 3 – 4 + 1 0 0 0 Check

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9-5 Solving Quadratic Equations by Factoring Additional Example 3: Application The height in feet of a diver above the water can be modeled by h(t) = –16t 2 + 8t + 8, where t is time in seconds after the diver jumps off a platform. Find the time it takes for the diver to reach the water. h = –16t 2 + 8t + 8 0 = –16t 2 + 8t + 8 0 = –8(2t 2 – t – 1) 0 = –8(2t + 1)(t – 1) The diver reaches the water when h = 0. Factor out the GCF, –8. Factor the trinomial.

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9-5 Solving Quadratic Equations by Factoring Additional Example 3 Continued –8 ≠ 0, 2t + 1 = 0 or t – 1= 0 Use the Zero Product Property. 2t = –1 or t = 1 Solve each equation. It takes the diver 1 second to reach the water. Check 0 = –16t 2 + 8t + 8 Substitute 1 into the original equation. 0 –16(1) 2 + 8(1) + 8 0 –16 + 8 + 8 0 Since time cannot be negative, does not make sense in this situation.

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9-5 Solving Quadratic Equations by Factoring Check It Out! Example 3 What if…? The equation for the height above the water for another diver can be modeled by h = –16t 2 + 8t + 24. Find the time it takes this diver to reach the water. h = –16t 2 + 8t + 24 0 = –16t 2 + 8t + 24 0 = –8(2t 2 – t – 3) 0 = –8(2t – 3)(t + 1) The diver reaches the water when h = 0. Factor out the GCF, –8. Factor the trinomial.

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9-5 Solving Quadratic Equations by Factoring –8 ≠ 0, 2t – 3 = 0 or t + 1= 0 Use the Zero Product Property. 2t = 3 or t = –1 Solve each equation. Since time cannot be negative, –1 does not make sense in this situation. It takes the diver 1.5 seconds to reach the water. Check 0 = –16t 2 + 8t + 24 Substitute 1.5 into the original equation. 0 –16(1.5) 2 + 8(1.5) + 24 0 –36 + 12 + 24 0 Check It Out! Example 3 Continued t = 1.5

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9-5 Solving Quadratic Equations by Factoring Lesson Quiz: Part I Use the Zero Product Property to solve each equation. Check your answers. 1. (x – 10)(x + 5) = 0 2. (x + 5)(x) = 0 Solve each quadratic equation by factoring. Check your answer. 3. x 2 + 16x + 48 = 0 4. x 2 – 11x = –24 10, –5 –5, 0 –4, –12 3, 8

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9-5 Solving Quadratic Equations by Factoring Lesson Quiz: Part II 1, –7 –9 –2 5 s 5. 2x 2 + 12x – 14 = 0 6. x 2 + 18x + 81 = 0 7. –4x 2 = 16x + 16 8. The height of a rocket launched upward from a 160 foot cliff is modeled by the function h(t) = –16t 2 + 48t + 160, where h is height in feet and t is time in seconds. Find the time it takes the rocket to reach the ground at the bottom of the cliff.

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