Problem 40 - College Algebra

Question

Add or subtract as indicated. $$\frac{x^{2}-4 x}{x^{2}-x-6}-\frac{x-6}{x^{2}-x-6}$$

Step-by-Step Solution

Answer

The result of the subtraction is \( \frac{x-3}{x+3} \).

Step 1: Identifying like terms

In the given exercise, the denominators of both fractions are the same: \(x^{2}-x-6\). The numerators are \(x^{2}-4x\) and \(x-6\). The aim is to subtract the second fraction from the first.

Step 2: Subtracting fractions

Since the denominators are the same, the fractions can be subtracted by subtracting the numerators. This gives: \(\frac{x^{2}-4x-(x-6)}{x^{2}-x-6}\).

Step 3: Simplifying the Fraction

Distribute the negative sign in the numerator: \(\frac{x^{2}-4x-x+6}{x^{2}-x-6}\), and combine like terms to give: \(\frac{x^{2}-5x+6}{x^{2}-x-6}\).

Step 4: Factoring the numerator and denominator

Factoring the numerator and denominator results in: \(\frac{(x-2)(x-3)}{(x-2)(x+3)}\).

Step 5: Simplifying the Fraction

The term (x-2) is in both the numerator and the denominator, so it cancels out, leaving the simplified fraction as: \( \frac{x-3}{x+3}\).