Problem 40 - College Algebra Essentials

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 operation is \(\frac{x^{2}-5x+6}{x^{2}-x-6}\).

Step 1 Identify common denominator

The task involves subtracting two fractions. It can be simplified because both fractions share the same denominator \((x^{2}-x-6)\). Hence, we can combine them into one fraction and perform the subtraction operation in the numerator.

Step 2 Combine fractions into one

Knowing the fractions share the common denominator, combine them into one fraction. The new fraction is \(\frac{(x^{2}-4x)-(x-6)}{x^{2}-x-6}\).

Step 3 Simplify the numerator

Now, simplify the numerator by performing the subtraction operation. During the subtraction, be careful of the minus sign. Thus, the simplified fraction after performing the subtraction in the numerator becomes \(\frac{x^{2}-4x-x+6}{x^{2}-x-6}\).

Step 4 Simplify the numerator further

Simplify the numerator further, by combining like terms. Thus, the simplified fraction becomes \(\frac{x^{2}-5x+6}{x^{2}-x-6}\).