Solving Equations by Factoring
Factoring is a method that can be used to solve equations of a degree higher than 1. This method uses the zero product rule.
If ( a)( b) = 0, then
Either ( a) = 0, ( b) = 0, or both.
Example 1
Solve x( x + 3) = 0.
x( x + 3) = 0
Apply the zero product rule.
Check the solution.
The solution is x = 0 or x = –3.
Example 2
Solve x 2 – 5 x + 6 = 0.
x 2 – 5 x + 6 = 0
Factor.
( x – 2)( x – 3) = 0
Apply the zero product rule.
The check is left to you. The solution is x = 2 or x = 3.
Example 3
Solve 3 x(2 x – 5) = –4(4 x – 3).
3 x(2 x – 5) = –4(4 x – 3)
Distribute.
6 x 2 – 15 x = –16 x + 12
Get all terms on one side, leaving zero on the other, in order to apply the zero product rule.
6 x 2 + x – 12 = 0
Factor.
(3 x – 4)(2 x + 3) = 0
Apply the zero product rule.
The check is left to you. The solution is or .
Example 4
Solve 2 y 3 = 162 y.
2 y 3 = 162 y
Get all terms on one side of the equation.
2 y 3 – 162 y = 0
Factor (GCF).
2 y( y 2 – 81) = 0
Continue to factor (difference of squares).
2 y( y + 9)( y – 9) = 0
Apply the zero product rule.
The check is left to you. The solution is y = 0 or y = –9 or y = 9.