Solving Quadratic Equations by Factoring
Quadratic Equation
If a, b, and c are real numbers, with a ≠ 0, then the equation
ax^{2} + bx + c = 0 is called a quadratic equation.
Keep the following strategy in mind
when solving equations by factoring.
Strategy for Solving Equations by Factoring
1. Write the equation with 0 on the righthand side.
2. Factor the lefthand side.
3. Use the zero factor property to get simpler equations. (Set each factor equal
to 0.)
4. Solve the simpler equations.
5. Check the answers in the original equation.
Example
Solving a quadratic equation by factoring
Solve each equation.
a) 10x^{2} = 5x
b) 3x  6x^{2} = 9
Solution
a) Use the steps in the strategy for solving equations by factoring:
10x^{2} = 5x 

Original equation 
10x^{2}  5x = 0 

Rewrite with zero on the righthand side. 
5x(2x  1) = 0 

Factor the lefthand side. 
5x = 0 
or 
2x  1 = 0 

Zero factor property 
x = 0 
or 


Solve for x.

The solution set is
. Check each solution in the original equation.
b) First rewrite the equation with 0 on the righthand side and the lefthand side in
order of descending exponents:
3x  6x^{2} = 9 

Original equation 
6x^{2} + 3x + 9 = 0 

Add 9 to each side. 
2x^{2}  x  3 = 0 

Divide each side by 3. 
(2x  3)(x + 1) 

Factor. 
2x  3 = 0 
or 
x + 1 = 0 

Zero factor property 

or 
x = 1 

Solve for x. 
The solution set is
. Check each solution in the original equation.
Caution
If we divide each side of 10x^{2} = 5x by 5x, we get 2x = 1, or
. We do not get x
= 0. By dividing by 5x we have lost one of the factors and
one of the solutions.
