Example 1A Solve the equation. Check your answer. (x – 7)(x + 2) = 0 Set each factor = 0 x – 7 = 0 or x + 2 = 0 Solve each equation. x = 7 or x = –2 The solutions are 7 and –2.

  Example 1A Continued (7 – 7)(7 + 2) 0 (0)(9) 0 Check (x – 7)(x + 2) = 0 (7 – 7)(7 + 2) 0 (0)(9) 0 0 0  Substitute each solution for x into the original equation. Check (x – 7)(x + 2) = 0 (–2 – 7)(–2 + 2) 0 (–9)(0) 0 0 0 

Substitute each solution for x into the original equation. Example 1B Solve each equation. Check your answer. x(x – 2) = 0 x(x – 2) = 0 Set each factor = 0 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.

Substitute each solution for x into the original equation. Check It Out! Example 1a Solve each equation. Check your answer. x(x + 4) = 0 Set each factor = 0 x = 0 or x + 4 = 0 Solve the second equation. x = –4 The solutions are 0 and –4. 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 0 Substitute each solution for x into the original equation. 

Check It Out! Example 1b Solve the equation. Check your answer. (3x + 4)(5x – 3) = 0 Set each factor = 0 3x + 4 = 0 or 5x – 3 = 0   Solve each equation.  

If a quadratic equation is written in standard form, ax2 + bx + c = 0, then to solve the equation, you may need to factor before using the Zero Product Property.

Example 2A: Finding Zeros by Factoring Find the zeros of the function by factoring. f(x) = x2 – 4x – 12 x2 – 4x – 12 = 0 Set the function equal to 0. (x + 2)(x – 6) = 0 Factor: Find factors of –12 that add to –4. x + 2 = 0 or x – 6 = 0 Set factors = 0 x= –2 or x = 6 Solve each equation.

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 Set each factor = 0 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 2C: 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 x = 6 or x = 6 Solve each equation. Both factors result in the same solution, so there is one solution, 6.

Find the zeros of the function by factoring. Check It Out! Example 2a Find the zeros of the function by factoring. f(x)= x2 – 5x – 6 x2 – 5x – 6 = 0 Set the function equal to 0. (x + 1)(x – 6) = 0 Factor: Find factors of –6 that add to –5. x + 1 = 0 or x – 6 = 0 x = –1 or x = 6 Solve each equation.

Find the zeros of the function by factoring. Check It Out! Example 2b Find the zeros of the function by factoring. g(x) = x2 – 8x x2 – 8x = 0 Set the function to equal to 0. Factor: Need factors of 0 that add to -8 x(x – 8) = 0 x = 0 or x – 8 = 0 x = 0 or x = 8 Solve each equation.

Example 2B: Finding Zeros by Factoring Find the zeros of the function by factoring. g(x) = 3x2 + 18x 3x2 + 18x = 0 Set the function to equal to 0.   Factor: Divide by 3, then find factors of 0 that add to 6 3x = 0 or x + 6 = 0 x = 0 or x = –6 Solve each equation.

Check It Out! Example 2a Solve the quadratic equation by factoring. Check your answer. x2 – 6x + 9 = 0 (x – 3)(x – 3) = 0 Factor the trinomial. x – 3 = 0 or x – 3 = 0 x = 3 or x = 3 Solve each equation. Both equations result in the same solution, so there is one solution, 3. x2 – 6x + 9 = 0 (3)2 – 6(3) + 9 0 9 – 18 + 9 0 0 0  Check Substitute 3 into the original equation.

Check It Out! Example 2d Solve the quadratic equation by factoring. Check your answer. 3x2 – 4x + 1 = 0 (3x – 1)(x – 1) = 0 Factor the trinomial. 3x – 1 = 0 or x – 1 = 0 or x = 1 Solve each equation. The solutions are and x = 1.

Lesson Quiz: Part I 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 10, –5 –5, 0 –4, –12

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