Problem 39 - College Algebra

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 result of the addition/subtraction is \(\frac{3x + 12}{x^{2}+x-12}\).

Step 1: Identify the common denominator

The common denominator for these two fractions is \(x^{2}+x-12\). This simplifies the process, and it means that you do not have to find the least common denominator since it is already provided.

Step 2: Subtract the numerators

Since the fractions have a common denominator, you can directly subtract the numerators. Do this by rewriting the expression to subtract the numerators from each other: \((x^{2}+3 x) - (x^{2} -12)\). This simplifies to \(3x + 12\).

Step 3: Write the simplified numerator over the common denominator

The simplified fraction after subtracting the numerators becomes \(\frac{3x + 12}{x^{2}+x-12}\).

Step 4: Simplify the fraction if possible

The fraction \(\frac{3x + 12}{x^{2}+x-12}\) cannot be simplified any further since the numerator and the denominator have no common factors.