To determine the excluded values of [tex]\( x \)[/tex] for the rational expression [tex]\(\frac{x^2 - 9x}{x^2 - 7x - 18}\)[/tex], we need to identify the values of [tex]\( x \)[/tex] that make the denominator equal to zero. These values are excluded because division by zero is undefined.
Let's proceed step by step:
1. Identify the denominator:
The denominator of the given expression is [tex]\( x^2 - 7x - 18 \)[/tex].
2. Find the values of [tex]\( x \)[/tex] that make the denominator zero:
We need to solve the equation [tex]\( x^2 - 7x - 18 = 0 \)[/tex].
3. Solve for [tex]\( x \)[/tex]:
To solve this quadratic equation, we can factor it. We look for two numbers that multiply to [tex]\(-18\)[/tex] and add to [tex]\(-7\)[/tex].
You can perform factoring by:
[tex]\(\quad x^2 - 7x - 18 = (x - 9)(x + 2) = 0\)[/tex]
4. Determine the solutions to the equation:
Set each factor equal to zero:
[tex]\[
x - 9 = 0 \quad \text{or} \quad x + 2 = 0
\][/tex]
Solving these, we get:
[tex]\[
x = 9 \quad \text{and} \quad x = -2
\][/tex]
5. Conclusion:
The critical points \{ [tex]\( x = 9 \)[/tex] and [tex]\( x = -2 \)[/tex] \} make the denominator zero and are therefore excluded values for [tex]\( x \)[/tex].
So, the excluded values of [tex]\( x \)[/tex] for the expression are:
[tex]\[ x = -2 \quad \text{and} \quad x = 9 \][/tex]
Therefore, the correct choice is:
[tex]\[ \boxed{x = -2, x = 9} \][/tex]
