Problem 39 - College Algebra Essentials

Question

Add or subtract as indicated. $$\frac{x^{2}+3 x}{x^{2}+x-12}-\frac{x^{2}-12}{x^{2}+x-12}$$

Step-by-Step Solution

Answer

The simple form of the expression is \(\frac{3x + 12}{x^{2}+x-12}\)

Step 1: Identify the Denominator

From the problem, it can be observed that both fractions have a common denominator: \(x^{2}+x-12\). This is useful because when adding or subtracting fractions with the same denominator, the denominator stays the same. The operation works on the numerator only.

Step 2: Subtract Numerators

Since the denominators of the two fractions are the same, we can directly subtract the numerators. The subtraction of the numerators, \((x^{2}+3x)-(x^{2}-12)\), simplifies first to \(x^{2}+3x - x^{2} + 12\), which can be further simplified to \(3x + 12\).

Step 3: Final Result

Substituting the result of subtracting the numerators into one of the fractions gives the result: \(\frac{3x + 12}{x^{2}+x-12}\)