Question: (\[ \frac{x^2 - 2x - 24}{x^2 - 16} \]\)…

\(\[ \frac{x^2 - 2x - 24}{x^2 - 16} \]\)

Solution

To simplify the expression \(\frac{x^2 - 2x - 24}{x^2 - 16}\), we will factor both the numerator and the denominator. First, factor the numerator \(x^2 - 2x - 24\). Find two numbers that multiply to \(-24\) and add to \(-2\). These numbers are \(-6\) and \(4\). Rewrite the quadratic: \[ x^2 - 2x - 24 = (x - 6)(x + 4) \] Next, factor the denominator \(x^2 - 16\). This is a difference of squares: \[ x^2 - 16 = (x - 4)(x + 4) \] Now, substitute the factored forms back into the expression: \[ \frac{(x - 6)(x + 4)}{(x - 4)(x + 4)} \] Cancel the common term \((x + 4)\) from the numerator and the denominator: \[ \frac{x - 6}{x - 4} \] The simplified expression is: \[ \frac{x - 6}{x - 4} \] Note that \(x \neq 4\) and \(x \neq -4\) due to the original denominator, to prevent division by zero.