Solving Equations by Factoring
Objective Learn to solve equations that are
in factored form.
This lesson begins with a review of factoring, and then uses
factored trinomials to solve equations. This technique
(factoring) is very useful in problem solving.
Factoring Expressions
The following are good examples.
Example 1
Factor x^{ 2}  4x  21.
Solution
Since the coefficient of x^{ 2} is 1, factor this
expression as ( x + a )( x + b ) for some integer values of a and
b. Now, ( x + a ) and ( x + b ) are multiplied to get x^{ 2}
+ ( a + b )x + ab , which is supposed to equal x^{ 2} 
4x  21. Comparing x^{ 2} + ( a + b )x + ab and x^{ 2}
 4x  21, we see that the sum a + b = 4 and the product ab =
21. So, the first step is to find the integral factors of  21,
for possible choices of a and b .
 21 = 21 Â· 1 
 21 = 21 Â· (1) 
 21 = 3 Â· 7 
 21 = 3 Â· (7) 
Look at the above factor pairs of 21 to see if any of them
have a sum equal to  4. If a = 3 and b = 7, then a + b = 4.
The factorization is given below.
x^{ 2}  4x  21 
= ( x + a )( x + b ) 


= ( x + 3 )[ x + (7)] 
a = 3, b = 7 

= ( x + 3 )( x  7 ) 

You should multiply ( x + 3) and ( x  7) to check that the
factorization is correct.
Example 2
Factor 2x^{ 3}  3x^{ 2}  2x .
Solution
There is a common factor of x in each of the terms. So begin
by factoring out x.
2x^{ 3}  3x^{ 2}  2x = x( 2x^{ 2} 
3x  2 )
The next step is to factor 2x^{ 2}  3x  2. That is,
write this expression as ( ax + b )( cx + d ) for some values of
a, b, c, and d. When we multiply this out we get acx^{ 2}
+ ( ad + bc )x + bd , which is supposed to equal 2x^{ 2}
 3x  2. So, ac = 2. Since 2 only has factors 2 and 1, let a = 2
and c = 1. We then have to find b and d. Since the coefficient of
x , ad + bc , is now 2d + b Â· 1 or 2d + b , 2d + b = 3 and bd =
2. The factors of  2 are 2 and 1, so b and d must be 2 or 1. On the other hand,
2d + b = 3. By checking the possibilities, we see that we must
have d = 2 and b = 1. Substitute these values for a, b, c, and
d.
2x^{ 3}  3x^{ 2}
 2x 
= ( ax + b )( cx + d ) 


= ( 2x + 1 )( x  2 ) 
a = 2, b = 1, c = 1, d = 2 
So,
2x^{ 3}  3x^{ 2}
 2x 
= x( 2x^{ 2}  3x  2 ) 

= x( 2x + 1 )( x  2 ) 
Again, you should multiply out x( 2x + 1 )( x  2 ) to check
the factoring. 