To determine the locations of the vertical asymptotes for the function
[tex]\[
f(x) = \frac{7}{x^2 - 2x - 24}
\][/tex]
we need to find the values of [tex]\( x \)[/tex] where the denominator equals zero. Vertical asymptotes occur at these points because the function approaches infinity or negative infinity.
### Step-by-Step Solution:
1. Identify the denominator of the function:
[tex]\[
x^2 - 2x - 24
\][/tex]
2. Set the denominator equal to zero to find where the function is undefined:
[tex]\[
x^2 - 2x - 24 = 0
\][/tex]
3. Solve the quadratic equation by factoring. We need to find two numbers that multiply together to give [tex]\(-24\)[/tex] and add together to give [tex]\(-2\)[/tex].
After examining various combinations, we see that [tex]\((x - 6)(x + 4) = 0\)[/tex] works because:
[tex]\[
\begin{cases}
-6 \cdot 4 = -24 \quad (\text{product}) \\
-6 + 4 = -2 \quad (\text{sums})
\end{cases}
\][/tex]
4. Set each factor equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
\begin{aligned}
x - 6 &= 0 \quad \Rightarrow \quad x = 6 \\
x + 4 &= 0 \quad \Rightarrow \quad x = -4
\end{aligned}
\][/tex]
5. Conclusion: The vertical asymptotes are at [tex]\( x = 6 \)[/tex] and [tex]\( x = -4 \)[/tex].
Therefore, the locations of the vertical asymptotes for [tex]\( f(x) = \frac{7}{x^2 - 2x - 24} \)[/tex] are [tex]\( x = 6 \)[/tex] and [tex]\( x = -4 \)[/tex].
The correct answer is:
[tex]\[
x = 6 \text{ and } x = -4
\][/tex]
