The fraction `(x-2)/(x^(2)+4x-21)` is not defined when x=

Options

A. `2`
B. `7 and -3`
C. `-7 or 3`
D. `-7 or -3`

Answer

To determine when the fraction \(\frac{x-2}{x^2 + 4x - 21}\) is not defined, we need to find the values of \(x\) that make the denominator equal to zero.

### Step 1: Set the denominator equal to zero
We start with the denominator:
\[ x^2 + 4x - 21 = 0 \]

### Step 2: Factor the quadratic equation
Next, we will factor the quadratic expression. We look for two numbers that multiply to \(-21\) (the constant term) and add to \(4\) (the coefficient of \(x\)). The numbers \(7\) and \(-3\) satisfy this condition.

Thus, we can factor the equation as:
\[ (x + 7)(x - 3) = 0 \]

### Step 3: Solve for \(x\)
Now, we set each factor equal to zero:
1. \(x + 7 = 0\) → \(x = -7\)
2. \(x - 3 = 0\) → \(x = 3\)

### Conclusion
The values of \(x\) that make the denominator zero (and thus the fraction undefined) are:
\[ x = -7 \quad \text{and} \quad x = 3 \]

### Final Answer
The fraction \(\frac{x-2}{x^2 + 4x - 21}\) is not defined when \(x = -7\) and \(x = 3\).

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