Warm Up: Factor each expression    

Zero Product Property: If the product of two quantities equals zero, at least one of the quantities equals 0.

Example 1: Use the Zero Product Property Use the Zero Product Property to solve the equation. Check your answer. (x – 7)(x + 2) = 0 Use the Zero Product Property. x – 7 = 0 or x + 2 = 0 Solve each equation. x = 7 or x = –2 The solutions are 7 and –2.

  Example 2: Use the Zero Product Property Use the Zero Product Property to solve each equation. Check your answer. x(x – 2) = 0 (x)(x – 2) = 0 Use the Zero Product Property. x = 0 or x – 2 = 0 Solve the second equation. x = 2 The solutions are 0 and 2. (x – 2)(x) = 0 (2 – 2)(2) 0 (0)(2) 0 0 0  Check (x – 2)(x) = 0 (0 – 2)(0) 0 (–2)(0) 0 0 0  Substitute each solution for x into the original equation.

Check It Out! Example 1b Use the Zero Product Property to solve the equation. Check your answer. (3x + 5)(x – 3) = 0 Use the Zero Product Property. 3x + 5 = 0 or x – 3 = 0   Solve each equation.  

Steps to solve quadratics: Factor the quadratic. Set each part equal to 0. Solve each part.

Example 3: 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 The solutions are 4 and 2. Solve each equation. x2 – 6x + 8 = 0 (4)2 – 6(4) + 8 0 16 – 24 + 8 0 0 0  Check x2 – 6x + 8 = 0 (2)2 – 6(2) + 8 0 4 – 12 + 8 0 0 0  Check

Example 4: Solving Quadratic Equations by Factoring Solve the quadratic equation by factoring. Check your answer. x2 + 4x = 21 The equation must be written in standard form. So subtract 21 from both sides. x2 + 4x = 21 –21 –21 x2 + 4x – 21 = 0 (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 The solutions are –7 and 3. Solve each equation.

Example 5: Solving Quadratic Equations by Factoring Solve the quadratic equation by factoring. Check your answer. x2 – 12x + 36 = 0 (x – 6)(x – 6) = 0 Factor the trinomial. x – 6 = 0 or x – 6 = 0 Use the Zero Product Property. x = 6 or x = 6 Solve each equation. Both factors result in the same solution, so there is one solution, 6.

Example 6: Solving Quadratic Equations by Factoring Solve the quadratic equation by factoring. Check your answer. –2x2 = 20x + 50 +2x2 +2x2 0 = 2x2 + 20x + 50 –2x2 = 20x + 50 The equation must be written in standard form. So add 2x2 to both sides. 2x2 + 20x + 50 = 0 Factor out the GCF 2. 2(x2 + 10x + 25) = 0 Factor the trinomial. 2(x + 5)(x + 5) = 0 2 ≠ 0 or x + 5 = 0 Use the Zero Product Property. x = –5 Solve the equation.

(x – 3)(x – 3) is a perfect square (x – 3)(x – 3) is a perfect square. Since both factors are the same, you solve only one of them. Helpful Hint

Solve the quadratic equation by factoring. Check your answer. Example 7 Solve the quadratic equation by factoring. Check your answer. 30x = –9x2 – 25 –9x2 – 30x – 25 = 0 Write the equation in standard form. –1(9x2 + 30x + 25) = 0 Factor out the GCF, –1. –1(3x + 5)(3x + 5) = 0 Factor the trinomial. –1 ≠ 0 or 3x + 5 = 0 Use the Zero Product Property. – 1 cannot equal 0. Solve the remaining equation.

Example 8 Solve the quadratic equation by factoring. Check your answer. 3x2 = 4x - 1 (3x – 1)(x – 1) = 0 Factor the trinomial. 3x – 1 = 0 or x – 1 = 0 Use the Zero Product Property. or x = 1 Solve each equation. The solutions are and x = 1.

Example 9 Example 10 Example 11      

Example 12: Application The 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 The diver reaches the water when h = 0. 0 = –16t2 + 8t + 8 0 = –8(2t2 – t – 1) Factor out the GFC, –8. 0 = –8(2t + 1)(t – 1) Factor the trinomial.

  Example 12 Continued Use the Zero Product Property. –8 ≠ 0, 2t + 1 = 0 or t – 1= 0 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 0 –16 + 8 + 8 0 0 Substitute 1 into the original equation. 

Example 13 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. When did the rocket reach its maximum height? What was the rocket’s maximum height? Find the time it takes the rocket to reach the ground at the bottom of the cliff.

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. x2 + 16x + 48 = 0 4. x2 – 11x = –24 10, –5 –5, 0 –4, –12 3, 8

Lesson Quiz: Part II 5. 2x2 + 12x – 14 = 0 1, –7 6. x2 + 18x + 81 = 0 –9 7. –4x2 = 16x + 16 –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. 5 s