Warm Up
Find 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.
(x + 5)(x + 7)
4. x2 + 12x + 35
(x – 7)(x + 9)
5. x2 + 2x – 63
6. x2 – 10x + 16
(x – 2)(x – 8)
7. 2x2 – 16x + 32
2(x – 4)2
Solving
Polynomials By
Factoring
Objective
I will solve quadratic equations by
factoring.
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
The solutions are 7 and –2.
Use the Zero Product
Property.
Solve each equation.
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)(9)
0
0
0
0
Check (x – 7)(x + 2) = 0
(–2 – 7)(–2 + 2)
(–9)(0)
0
0
0
0
Substitute each solution
for x into the original
equation.
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
The solutions are 0 and 2.
Check (x – 2)(x) = 0
(0 – 2)(0)
(–2)(0)
0
0
0
0
Use the Zero Product
Property.
Solve the second
equation.
(x – 2)(x) = 0
Substitute each (2 – 2)(2)
solution for x into
(0)(2)
the original
0
equation.
0
0
0
Check It Out! Example 1a
Use the Zero Product Property to solve each
equation. Check your answer.
(x)(x + 4) = 0
(x + 4)(x – 3) = 0
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.
Use the Zero Product
Property.
Solve each equation.
x – 4 = 0 or x – 2 = 0
x = 4 or x = 2
The solutions are 4 and 2.
Check
Check
x2 – 6x + 8 = 0
x2 – 6x + 8 =
(4)2 – 6(4) + 8 0
(2)2 – 6(2) + 8
16 – 24 + 8 0
4 – 12 + 8
0 0
0
0
0
0
0
Example 2B: Solving Quadratic Equations by Factoring
Solve the quadratic equation by factoring. Check
your answer.
x2 + 4x = 21
x2 + 4x = 21
–21 –21
x2 + 4x – 21 = 0
The equation must be written in
standard form. So subtract 21 from
both sides.
(x + 7)(x –3) = 0
Factor the trinomial.
x + 7 = 0 or x – 3 = 0
x = –7 or x = 3
Use the Zero Product Property.
Solve each equation.
The solutions are –7 and 3.
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
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.
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) = 0
2(x + 5)(x + 5) = 0
2≠0
or
x+5=0
x = –5
Factor the trinomial.
Use the Zero Product Property.
Solve the equation.
Example 2D Continued
Solve 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.
Helpful Hint
(x – 3)(x – 3) is a perfect square. Since both
factors are the same, you solve only one of
them.
Check It Out! Example 2a
Solve the quadratic equation by factoring.
Check your answer.
x2 – 6x + 9 = 0
x2 + 4x = 5
30x = –9x2 – 25
3x2 – 4x + 1 = 0
Lesson Quiz: Part I
Use 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 II
5. 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