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

2. Learning Target
Students will be able to: Solve quadratic equations by factoring.

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

4. Use the Zero Product Property to solve the equation.
(x – 7)(x + 2) = 0
x – 7 = 0 or x + 2 = 0
x = 7 or x = –2
(x + 4)(x – 3) = 0
The solutions are 7 and –2.
x + 4 = 0 or x – 3 = 0
x = –4 or x = 3
(x – 2)(x) = 0
The solutions are –4 and 3.
x = 0 or x – 2 = 0
x = 2
The solutions are 0 and 2.

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

6. Solve the quadratic equation by factoring.
x2 – 6x + 8 = 0
(x – 4)(x – 2) = 0
x – 4 = 0 or x – 2 = 0
x = 4 or x = 2
The solutions are 4 and 2.

7. x2 + 4x = 21
–21 –21
x2 + 4x – 21 = 0
x2 + 4x = 21
(x + 7)(x –3) = 0
x + 7 = 0 or x – 3 = 0
x = –7 or x = 3
The solutions are –7 and 3.

8. Solve the quadratic equation by factoring.
–2x2 = 20x + 50
+2x x2
0 = 2x2 + 20x + 50
–2x2 = 20x + 50
0 = 2(x2 + 10x + 25)
0 = 2(x + 5)(x + 5)
x + 5 = 0
x = –5

9. (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

10. Solve the quadratic equation by factoring. Check your answer.
x2 – 6x + 9 = 0
(x – 3)(x – 3) = 0
x – 3 = 0
x = 3

11. Solve the quadratic equation by factoring.
x2 + 4x = 5
–5 –5
x2 + 4x – 5 = 0
x2 + 4x = 5
(x – 1)(x + 5) = 0
x – 1 = 0 or x + 5 = 0
x = or x = –5
The solutions are 1 and –5.

12. Solve the quadratic equation by factoring.
30x = –9x2 – 25
9x2 + 30x + 25 = 0
(3x + 5)(3x + 5) = 0
3x + 5 = 0

13. 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
2t + 1 = 0 or t – 1= 0
0 = –16t2 + 8t + 8
2t = –1 or t = 1
0 = –8(2t2 – t – 1) 
It takes the diver 1 second to reach the water.

14. HW pp /20-44,46-50,56-67