The problem asks to find the value(s) of $x$ for which the expression $\frac{x^2 - 7x + 12}{x^2 - 9x + 18}$ is undefined. A rational expression is undefined when the denominator is equal to zero. Therefore, we need to find the values of $x$ that satisfy the equation $x^2 - 9x + 18 = 0$.

1. Problem Description

The problem asks to find the value(s) of $x$ for which the expression $\frac{x^2 - 7x + 12}{x^2 - 9x + 18}$ is undefined. A rational expression is undefined when the denominator is equal to zero. Therefore, we need to find the values of $x$ that satisfy the equation $x^2 - 9x + 18 = 0$.

2. Solution Steps

We need to solve the quadratic equation $x^2 - 9x + 18 = 0$. We can solve this by factoring.

We look for two numbers that multiply to 18 and add up to -

9. These two numbers are -3 and -

6. So we can factor the quadratic expression as follows:

$x^2 - 9x + 18 = (x - 3)(x - 6)$

Now we set the factored expression equal to zero:

$(x - 3)(x - 6) = 0$

This equation is satisfied if either $x - 3 = 0$ or $x - 6 = 0$.

Solving for $x$ in each case:

$x - 3 = 0 \implies x = 3$

$x - 6 = 0 \implies x = 6$

Therefore, the expression is undefined when $x = 3$ or $x = 6$.

3. Final Answer

The expression is undefined for $x = 3$ and $x = 6$.