Solving Quadratic Equations by Factoring
Quadratic Equation
If a, b, and c are real numbers, with a ≠ 0, then the equation
ax2 + 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 right-hand side.
2. Factor the left-hand 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) 10x2 = 5x
b) 3x - 6x2 = -9
Solution
a) Use the steps in the strategy for solving equations by factoring:
10x2 = 5x |
|
Original equation |
10x2 - 5x = 0 |
|
Rewrite with zero on the right-hand side. |
5x(2x - 1) = 0 |
|
Factor the left-hand 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 right-hand side and the left-hand side in
order of descending exponents:
3x - 6x2 = -9 |
|
Original equation |
-6x2 + 3x + 9 = 0 |
|
Add 9 to each side. |
2x2 - 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 10x2 = 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.
|