Problem 54 - College Algebra

Question

Add or subtract as indicated. $$\frac{x}{x^{2}-2 x-24}-\frac{x}{x^{2}-7 x+6}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{-5x}{(x - 6)(x + 4)(x - 1)} \).

Step 1: Factoring the Denominators

Factorize both denominators to find the roots and help us determine the common denominator. The factorization of first denominator \(x^{2}-2 x-24\) is \((x -6)(x +4)\) and for the second denominator \(x^{2}-7 x+6\) is \((x - 1)(x - 6)\).

Step 2: Determining the Common Denominator

As we can observe, \(x - 6\) is common in both factorized denominators, so the remaining different factors will be part of the common denominator. Therefore, the common denominator is \((x - 6)(x - 1)(x + 4)\).

Step 3: Rewrite Fractions with the Common Denominator

Rewrite both fractions in terms of the common denominator by multiplying the numerator and the denominator of the first fraction by \((x - 1)\), and the numerator and the denominator of the second fraction by \((x + 4)\), to get: \(\frac{x(x - 1)}{(x - 6)(x + 4)(x - 1)} - \frac{x(x + 4)}{(x - 6)(x - 1)(x + 4)}\).

Step 4: Simplifying the Expression

Simplify the expression by subtracting the numerators and divide by the common denominator to get \(\frac{x(x - 1)- x(x + 4)}{(x - 6)(x + 4)(x - 1)}. To further simplify, expand the terms in the numerator to get \(\frac{x^2 - x - x^2 -4x}{(x - 6)(x + 4)(x - 1)} = \frac{-5x}{(x - 6)(x + 4)(x - 1)}.\)