Questions: Identify the domain of the following rational expression: (x+4)/(x^2+x-12)

Identify the domain of the following rational expression: (x+4)/(x^2+x-12)
Transcript text: Identify the domain of the following rational expression: $\frac{x+4}{x^{2}+x-12}$
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Solution

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Solution Steps

Step 1: Identify the Denominator

The given rational expression is

\[ \frac{x+4}{x^{2}+x-12} \]

The domain of a rational expression is all real numbers except those that make the denominator zero. Therefore, we need to find the values of \(x\) that make the denominator zero.

Step 2: Solve for the Denominator

Set the denominator equal to zero and solve for \(x\):

\[ x^2 + x - 12 = 0 \]

Step 3: Factor the Quadratic

Factor the quadratic equation:

\[ x^2 + x - 12 = (x - 3)(x + 4) = 0 \]

Step 4: Find the Values that Make the Denominator Zero

Set each factor equal to zero:

  1. \(x - 3 = 0 \Rightarrow x = 3\)
  2. \(x + 4 = 0 \Rightarrow x = -4\)

These are the values that make the denominator zero.

Final Answer

The domain of the rational expression is all real numbers except \(x = 3\) and \(x = -4\). Therefore, the correct choice is:

\[ \boxed{\text{All real numbers except -4 and 3.}} \]

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