Solving Equations by Factoring
Using Factoring to Solve Equations
How is factoring useful in solving equations? Consider, for
example the factored equation
(3x  2)( x + 1) = 0.
To solve this, use the following important principle.
Zero Product Property
For all numbers a and b, if ab = 0, then a = 0, b = 0, or both
a and b equal 0.
Since (3x  2)( x + 1) = 0, then the Zero Product Property
states that either 3x  2 = 0, x + 1 = 0, or both equal zero.
3x  2 = 0 
or 
x + 1 = 0 
3x = 2 

x = 1 



So, the solutions of (3x  2)( x + 1) = 0 are and x = 1. Now apply
this property to solve an equation that must be factored first.
Example 1
Find the solutions of 2x^{ 3} 2x^{ 2} = 12x .
Solution
First, write this equation so that it is set equal to zero.
2x^{ 3}  2x^{ 2}
 12x = 0 

x ( 2x^{ 2}  2x 
12 ) = 0 
Since x is a factor of all the terms,
factor it out. 
2x ( x^{ 2}  x  6
) = 0 
Since each coefficient is even, factor
out a 2. 
Now factor x^{ 2}  x  6 as a product ( x + a )( x +
b ) for some values of a and b.
( x + a )( x + b ) 
= x^{ 2} + ( a + b )x + ab 

= x^{ 2}  x  6 
In other words, a + b = 1 and ab = 6. By looking at the
factors of 6, we see that a = 2 and b = 3 satisfy these
requirements. So x^{ 2}  x  6 = ( x + 2)( x  3). (Check
to see if this is correct by multiplying it out.)
2x ( x^{ 2}  x  6
) = 0 

2x( x + 2)( x  3) = 0 
Substitution 
Use the Zero Product Property to find the solutions.
2x = 0 
x + 2 = 0 
x  3 = 0 
x = 0 
x = 2 
x = 3 
So, the solutions of the equation are x = 0, x = 2, and x =
3.
Finally, point out that the solutions of the equation are the
roots (zeros) of the function
y = 2x^{ 3}  2x^{ 2}  12x.
Graphically, the roots are where the graph of the function y =
2x^{ 3}  2x^{ 2}  12x intersects the x axis. Plot
some points or use a graphing calculator to graph this function.
The roots are at x = 0, x = 2, and x = 3, as shown on the
figuer below.
