1. Objective
Solve quadratic equations by factoring.

2. 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.

3. Example 1A: 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.

4. Example 1A Continued
Use the Zero Product Property to solve the equation. Check your answer.
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 

5.   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
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 
Check (x – 2)(x) = 0
(0 – 2)(0) 0
(–2)(0) 0 
Substitute each solution for x into the original equation.

6. 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.

7. 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
The solutions are 4 and 2.
Solve each equation.
x2 – 6x + 8 = 0
(4)2 – 6(4)
16 –
0 0 
Check
x2 – 6x + 8 = 0
(2)2 – 6(2)
4 –
0 0 
Check

8. 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 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.

9. 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
Use the Zero Product Property.
x = or x = 6
Solve each equation.
Both factors result in the same solution, so there is one solution, 6.

10. Solve the quadratic equation by factoring. Check your answer.
Example 2C Continued
Solve the quadratic equation by factoring. Check your answer.
x2 – 12x + 36 = 0
Check Graph the related quadratic function. ●
The graph of y = x2 – 12x + 36 shows that one zero appears to be 6, the same as the solution from factoring. 

11. Example 2D: Solving Quadratic Equations by Factoring
Solve the quadratic equation by factoring. Check your answer.
–2x2 = 20x + 50
+2x x2
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.

12. Example 2D Continued
Solve the quadratic equation by factoring. Check your answer.
–2x2 = 20x + 50
Check
–2x2 = 20x + 50
–2(–5) (–5) + 50
– –
– –50 
Substitute –5 into the original equation.

13. Example 3: 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.

14.   Example 3 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 –
Substitute 1 into the original equation. 