For what values of x is (x ^ 2 + x - 12)/(x ^ 2 + 7x + 12) undefined?

The expression \(\frac{x^2 + x - 12}{x^2 + 7x + 12}\) is undefined when the denominator equals zero. So, let's first find when the denominator \(x^2 + 7x + 12 = 0\).

We will solve this quadratic equation by factoring.

### Step 1: Factor the denominator
We need two numbers that multiply to 12 (the constant term) and add to 7 (the coefficient of \(x\)).

Those numbers are 3 and 4 because:
\[
3 \times 4 = 12 \quad \text{and} \quad 3 + 4 = 7
\]

Thus, we can factor the quadratic as:
\[
x^2 + 7x + 12 = (x + 3)(x + 4)
\]

### Step 2: Set the factors equal to zero
Now, set each factor equal to zero to find the values of \(x\) that make the denominator zero.

\[
x + 3 = 0 \quad \Rightarrow \quad x = -3
\]
\[
x + 4 = 0 \quad \Rightarrow \quad x = -4
\]

### Conclusion:
The expression is undefined for \(x = -3\) and \(x = -4\).

Would you like more details on how this factoring works or have any other questions?

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Here are 5 related questions to expand your understanding:

1. How would the numerator of the given expression factor, and does it affect where the expression is undefined?
2. What is the domain of the function represented by the given expression?
3. How do you determine when a rational expression is undefined in general?
4. Can you describe how the behavior of the function changes near the undefined points \(x = -3\) and \(x = -4\)?
5. What happens to the graph of a rational function at points where it is undefined?

**Tip:** Whenever dealing with rational expressions, always check where the denominator equals zero to identify points of discontinuity.

For what values of x is (x^2 + x - 12)/(x^2 + 7x + 12) undefined?

Explore how to determine the values of x that make the rational expression (x^2 + x - 12)/(x^2 + 7x + 12) undefined by factoring the quadratic denominator. This solution highlights algebraic concepts like factoring quadratics and finding discontinuities. Suitable for students in Grades 8-10.

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