Question

simplify the following expression completely. \\(\frac{x^2 - 3x - 54}{x^2 - 7x - 18}\\)

Explanation:

Step1: Factor numerator and denominator

Factor \(x^2 - 3x - 54\): find two numbers that multiply to \(-54\) and add to \(-3\). These numbers are \(-9\) and \(6\). So, \(x^2 - 3x - 54=(x - 9)(x + 6)\).

Factor \(x^2 - 7x - 18\): find two numbers that multiply to \(-18\) and add to \(-7\). These numbers are \(-9\) and \(2\). So, \(x^2 - 7x - 18=(x - 9)(x + 2)\).

Step2: Cancel common factors

The expression becomes \(\frac{(x - 9)(x + 6)}{(x - 9)(x + 2)}\). Cancel the common factor \((x - 9)\) (assuming \(x
eq9\) to avoid division by zero).

Answer:

\(\frac{x + 6}{x + 2}\)