Question: (\[ \frac{x^2 - 7x + 12}{x^2 - x - 12} \]\)…

\(\[ \frac{x^2 - 7x + 12}{x^2 - x - 12} \]\)

Solution

To simplify the expression \[ \frac{x^2 - 7x + 12}{x^2 - x - 12} \] we need to factor both the numerator and the denominator. First, factor the numerator \(x^2 - 7x + 12\). Find two numbers that multiply to \(12\) and add to \(-7\). These numbers are \(-3\) and \(-4\). So, the numerator factors as: \[ x^2 - 7x + 12 = (x - 3)(x - 4) \] Next, factor the denominator \(x^2 - x - 12\). Find two numbers that multiply to \(-12\) and add to \(-1\). These numbers are \(-4\) and \(3\). So, the denominator factors as: \[ x^2 - x - 12 = (x - 4)(x + 3) \] Now substitute the factored forms back into the original expression: \[ \frac{(x - 3)(x - 4)}{(x - 4)(x + 3)} \] Cancel out the common factor \((x - 4)\) from the numerator and the denominator: \[ \frac{x - 3}{x + 3} \] The simplified expression is \[ \frac{x - 3}{x + 3} \]