Solving Equations by Factoring
Definition: Any equation that can be put in the form ax^{2} + bx + c = 0 where a, b, and
c are real numbers (a ≠ 0), is called a quadratic equation. The
standard form for a
quadratic equation:
ax^{2} + bx + c = 0
ZeroFactor Theorem: For real numbers A and B, if AÂ·B = 0 then A = 0 or B = 0, or both.
WHAT TO DO: 
HOW TO DO IT: 
1. Solve the equation x^{ 2} − 5x − 6 = 0
Scratch: Find a numbers whose product is 6
with difference of 5 , larger sign â€œ−â€.
Set each factor equal to zero:
The solution:

x
^{2} − 5x − 6 = 0
(x − 6)(x + 1) = 0
(x − 6) = 0 or (x + 1) = 0
x = 6 or x = − 1

Check the equation x^{ 2}
− 5x − 6 = 0 
(6)^{2 }− 5(6) − 6 = 0 , (1)^{2} − 5(1) − 6 = 0 
2. Solve the equation x^{ 2} = 7x − 12
Rewrite as a standard equation:
Scratch: Find a pair of numbers whose
product is 12 with sum of 7.
Set each factor equal to zero:
The solution:

x^{ 2} = 7x − 12 x
^{2} − 7x + 12 = 0
(x − 4)(x − 3) = 0
(x − 4) = 0 or (x − 3) = 0
x = 4 or x = 3

Check the equation x^{ 2}
= 7x − 12 
(4)^{2}
= 7(4) − 12 , (3)^{2} = 7(3) − 12 
2. Solve the equation x^{ 2} = 12x
Rewrite as a standard equation:
Set each factor equal to zero:
The solution:

x
^{2} = 12x
x ^{2} − 12x = 0
xÂ·(x − 12) = 0
x = 0 or (x − 12) = 0
x = 0 or x = 12 
Check the equation x^{ 2} = 12x

(0)^{2} = 12(0) , (12)^{2} = 12(12) 
