Solving Quadratic Equations by Factoring
Solving Quadratic Equations by Factoring
A quadratic equation has the form:
ax2 + bx + c = 0
where a, b, and c are real numbers and a does not equal 0.
To solve a quadratic equation by using factoring, write the equation in standard form. This may involve manipulating the equation such that all terms are on one side, and zero is on the other. Next, factor the side of the equation that is not zero. Note that any number times zero equals zero, so either one factor is zero, or the other factor is zero in order to achieve a product of zero. Therefore, set each factor containing a variable equal to zero and solve each equation. Check each solution.
Example
Solve x2 + 6x = −8
Add 8 to both sides to put the equation in standard form.
x2 + 6x + 8 = 0
Factor the side of the equation that is not zero.
(x + ?)(x + ?) = 0
(Think: 2 × 4 = 8, and 2 + 4 = 6)
(x + 2)(x + 4) = 0
Set each factor equal to zero and solve both equations.
x + 2 = 0 OR x + 4 = 0
x = −2 OR x = −4
The solution set for the equation is {−4, −2}.
Practice
Solve the following equations.
1. x2 − 12x + 36 = 0
2. x2 = 9x + 22
3. x2 − 56 = x
4. 3x2 + 5x = 3x + 16
5. 5(x2 + 1) = 26x
Answers
1. 6
2. {−2, 11}
3. {−7, 8}
4. {, 2}
5. {, 5}