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9-5 Solving Quadratic Equations by Factoring

Warm UpFind each product. 1. (x + 2)(x + 7) 2. (x 11)(x + 5) x2 + 9x + 14 x2 6x 55 3. (x 10)2 x2 20x + 100 Factor each polynomial. 4. x2 + 12x + 35 (x + 5)(x + 7) 6. x2 10x + 16 (x 2)(x 8) 5. x2 + 2x 63 (x 7)(x + 9) 7. 2x2 16x + 32 2(x 4)2

You have solved quadratic equations by graphing. Another method used to solve quadratic equations is to factor and use the Zero Product Property.

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.

Additional Example 1A ContinuedUse the Zero Product Property to solve the equation. Check your answer.Check (x 7)(x + 2) = 0

(7 7)(7 + 2) (0)(9) 0 (2 7)(2 + 2) (9)(0) 0

0 0 0 0 0 0 Substitute each solution for x in the original equation.

Check (x 7)(x + 2) = 0

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)(x 2) = 0 x = 0 or x 2 = 0 x=2 Check (x 2)(x) = 0 (0 2)(0) (2)(0) 0 0 0 0 Use the Zero Product Property. Solve the second equation. Substitute (x 2)(x) = each solution (2 2)(2) for x in the (0)(2) original 0 equation. 0 0 0 0

Check It Out! Example 1aUse the Zero Product Property to solve each equation. Check your answer. (x)(x + 4) = 0 x = 0 or x + 4 = 0 x = 4 Check (x)(x + 4) = 0 (0)(0 + 4) (0)(4) 0 0 0 0 Use the Zero Product Property. Solve the second equation.

Substitute (x)(x +4) = 0 each solution for x in the (4)(4 + 4) 0 (4)(0) 0 original equation. 0 0

Check It Out! Example 1bUse 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.

Check It Out! Example 1b ContinuedUse the Zero Product Property to solve the equation. Check your answer. (x + 4)(x 3) = 0 Check (x + 4)(x 3 ) = 0 (4 + 4)(4 3) 0 (0)(7) 0 0 0 Check (x + 4)(x 3 ) = 0 (3 + 4)(3 3) (7)(0) 0 0 0 0 Substitute each solution for x in the original equation.

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.

Helpful Hint To review factoring techniques, see Lessons 8-3 through 8-5.

Additional Example 2A: Solving Quadratic Equations by Factoring Solve the quadratic equation by factoring. Check your answer. x2 6x + 8 = 0 (x 4)(x 2) = 0 Factor the trinomial. x 4 = 0 or x 2 = 0 Use the Zero Product Property. x = 4 or x = 2 Solve each equation. Check Check x2 6x + 8 = 0 x2 6x + 8 = 0 (4)2 6(4) + 8 0 (2)2 6(2) + 8 0 16 24 + 8 0 4 12 + 8 0 0 0 0 0

Additional Example 2B: Solving Quadratic Equations by Factoring Solve the quadratic equation by factoring. Check your answer. x2 + 4x = 21 The equation must be written in 2 + 4x = 21 x standard form. So subtract 21 21 21 from both sides. 2 + 4x 21 = 0 x(x + 7)(x 3) = 0 Factor the trinomial.

x + 7 = 0 or x 3 = 0 Use the Zero Product Property. x = 7 or x = 3 Solve each equation.

Additional Example 2B ContinuedCheck 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 = x2 + 4x 21 intersects the x-axis at x = 7 and x = 3, the same as the solutions from factoring.

Additional Example 2C: Solving Quadratic Equations by FactoringSolve the quadratic equation by factoring. Check your answer. x2 12x + 36 = 0 (x 6)(x 6) = 0 x 6 = 0 or x 6 = 0 x=6 or x=6 Factor the trinomial. Use the Zero Product Property. Solve each equation.

Both factors result in the same solution, so there is one solution, 6.

Additional Example 2C ContinuedCheck Graph the related quadratic function. The graph of y = x2 12x + 36 shows one zero at 6, the same as the solution from factoring.

Additional Example 2D: Solving Quadratic Equations by Factoring Solve the quadratic equation by factoring. Check your answer. 2x2 = 20x + 50 The equation must be written in 2x2 = 20x + 50 +2x2 +2x2 standard form. So add 2x2 to 0 = 2x2 + 20x + 50 both sides. 2x2 + 20x + 50 = 0 Factor out the GCF 2. 2(x2 + 10x + 25) = 02(x + 5)(x + 5) = 0 20 or Factor the trinomial.

x + 5 = 0 Use the Zero Product Property. x = 5 Solve the equation.

Additional Example 2D ContinuedSolve the quadratic equation by factoring. Check your answer. 2x2 = 20x + 50 Check 2x2 = 20x + 50 2(5)2 50 50 20(5) + 50 100 + 50 50 Substitute 5 into the original equation.

Check It Out! Example 2a Solve the quadratic equation by factoring. Check your answer.x2 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. Check x2 6x + 9 = (3)2 6(3) + 9 9 18 + 9 0 0 0 0 0 Factor the trinomial. Use the Zero Product Property. Solve each equation.

Substitute 3 into the original equation.

Check It Out! Example 2bSolve the quadratic equation by factoring. Check your answer. x2 + 4x = 5 x2 + 4x = 5 5 5 x2 + 4x 5 = 0 (x 1)(x + 5) = 0 x 1 = 0 or x + 5 = 0 x=1 or x = 5 Write the equation in standard form. Add 5 to both sides.

Factor the trinomial. Use the Zero Product Property. Solve each equation.

Check It Out! Example 2b ContinuedCheck 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 = x2 + 4x 5 intersects the x-axis at x = 1 and x = 5, the same as the solutions from factoring.

Check It Out! Example 2cSolve the quadratic equation by factoring. Check your answer. 30x = 9x2 25 9x2 30x 25 = 0 1(9x2 + 30x + 25) = 0 1(3x + 5)(3x + 5) = 0 1 0 or 3x + 5 = 0 Write the equation in standard form. Factor out the GCF, 1. Factor the trinomial. Use the Zero Product Property. 1 cannot equal 0. Solve the remaining equation.

Check It Out! Example 2c ContinuedCheck 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 = 9x2 30x 25 intersects the x-axis at x = , the same as the solution from factoring.

Check It Out! Example 2dSolve the quadratic equation by factoring. Check your answer. 3x2 4x + 1 = 0 (3x 1)(x 1) = 0 3x 1 = 0 or x 1 = 0 or x = 1 Factor the trinomial. Use the Zero Product Property. Solve each equation.

Check It Out! Example 2d ContinuedCheck 3x2 4x + 1 = 0 3 4 +1 0 Check 3x2 4x + 1 = 3(1)2 4(1) + 1 34+1 0

0 0 0 0

0

0

Additional Example 3: ApplicationThe height in feet of a diver above the water can be modeled by h(t) = 16t2 + 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 = 16t2 + 8t + 8 0= 16t2 + 8t + 8 The diver reaches the water when h = 0. Factor out the GCF, 8. Factor the trinomial.

0 = 8(2t2 t 1) 0 = 8(2t + 1)(t 1)

Additional Example 3 ContinuedUse the Zero Product 8 0, 2t + 1 = 0 or t 1= 0 Property. 2t = 1 or t = 1 Solve each equation. Since time cannot be negative, does not make sense in this situation.

It takes the diver 1 second to reach the water.

Check 0 = 16t2 + 8t + 8 0 16(1)2 + 8(1) + 8 Substitute 1 into the original equation. 0 16 + 8 + 8 0 0

Check It Out! Example 3What if? The equation for the height above the water for another diver can be modeled by h = 16t2 + 8t + 24. Find the time it takes this diver to reach the water. h = 16t2 + 8t + 24 0= 16t2 + 8t + 24 The diver reaches the water when h = 0. Factor out the GCF, 8. Factor the trinomial.

0 = 8(2t2 t 3) 0 = 8(2t 3)(t + 1)

Check It Out! Example 3 Continued8 0, 2t 3 = 0 or t + 1= 0 2t = 3 or t = 1 t = 1.5 It takes the diver 1.5 seconds to reach the water. Use the Zero Product Property. Solve each equation. Since time cannot be negative, 1 does not make sense in this situation.

Check 0 = 16t2 + 8t + 24 0 16(1.5)2 + 8(1.5) + 24 Substitute 1.5 into the 0 36 + 12 + 24 original equation. 0 0

Lesson Quiz: Part IUse the Zero Product Property to solve each equation. Check your answers. 1. (x 10)(x + 5) = 0 10, 5 2. (x + 5)(x) = 0 5, 0

Solve each quadratic equation by factoring. Check your answer. 3. x2 + 16x + 48 = 0 4, 12 4. x2 11x = 24 3, 8

Lesson Quiz: Part II5. 2x2 + 12x 14 = 0 1, 7 6. x2 + 18x + 81 = 0 7. 4x2 = 16x + 16 9 2

8. The height of a rocket launched upward from a 160 foot cliff is modeled by the function h(t) = 16t2 + 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. 5s