To solve the quadratic equation [tex]\( m^2 - 3m - 108 = 0 \)[/tex] by factoring, follow these steps:
1. Rewrite the equation: Start with the given equation:
[tex]\[
m^2 - 3m - 108 = 0
\][/tex]
2. Factor the quadratic expression: We need to find two numbers that multiply to give the constant term (-108) and add to give the coefficient of the linear term (-3). These two numbers are 12 and -9, because:
[tex]\[
12 \times (-9) = -108 \quad \text{and} \quad 12 + (-9) = 3
\][/tex]
Therefore, we can factor the quadratic expression as:
[tex]\[
(m - 12)(m + 9) = 0
\][/tex]
3. Solve for [tex]\( m \)[/tex]: Set each factor equal to zero and solve for [tex]\( m \)[/tex]:
- For [tex]\( m - 12 = 0 \)[/tex]:
[tex]\[
m - 12 = 0
\][/tex]
Add 12 to both sides to solve for [tex]\( m \)[/tex]:
[tex]\[
m = 12
\][/tex]
- For [tex]\( m + 9 = 0 \)[/tex]:
[tex]\[
m + 9 = 0
\][/tex]
Subtract 9 from both sides to solve for [tex]\( m \)[/tex]:
[tex]\[
m = -9
\][/tex]
4. State the solution: The values of [tex]\( m \)[/tex] that satisfy the equation [tex]\( m^2 - 3m - 108 = 0 \)[/tex] are:
[tex]\[
m = 12, -9
\][/tex]
