To determine the values of $x$ where the function $F(x) = \frac{x+2}{x^2 + 2x - 24}$ has vertical asymptotes, we need to analyze the denominator $x^2 + 2x - 24$.

Vertical asymptotes occur where the denominator is equal to zero, but the numerator is not zero at the same point.

Step 1: Factorize the denominator

We factorize $x^2 + 2x - 24$ as follows:

$x^2 + 2x - 24 = (x + 6)(x - 4)$

Step 2: Solve for $x$

The denominator equals zero when:

$(x + 6)(x - 4) = 0$

Setting each factor equal to zero:

$x + 6 = 0 \quad \text{or} \quad x - 4 = 0$

$x = -6 \quad \text{and} \quad x = 4$

Step 3: Check for vertical asymptotes

We need to confirm that the numerator $x + 2$ does not equal zero at these $x$-values. The numerator is $x + 2$, and it equals zero when:

$x + 2 = 0 \implies x = -2$

Thus:

• At $x = -6$ and $x = 4$, the denominator is zero, but the numerator is not zero. Therefore, vertical asymptotes occur at $x = -6$ and $x = 4$.

Final Answer:

The correct answers are:

• C. $x = 4$
• E. $x = -6$

Would you like further clarification on this concept or additional examples?

Related Questions:

1. What is the definition of a vertical asymptote?
2. How do you identify horizontal asymptotes of a rational function?
3. How do you factorize a quadratic expression like $x^2 + 2x - 24$?
4. What happens if the numerator and denominator both equal zero at the same $x$-value?
5. How do vertical asymptotes relate to undefined points on a graph?

Tip:

Always check for common factors between the numerator and denominator when finding vertical asymptotes. If there are common factors, the point might be a hole instead of an asymptote.