Problem 70 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x^{2}+9 x}{x^{2}-2 x-3}+\frac{5}{3-x}$$

Step-by-Step Solution

Answer

\(\frac{x^{2} + 9x + 5}{(x - 3)(x + 1)}\)

Step 1: Analyze Each Denominator

The first denominator is a quadratic equation \(x^{2}-2x-3\), which can be factored into \((x-3)(x+1)\). The second denominator is \(3 - x\), which can be rewritten as \(-(x - 3)\) to align with the factor from the first denominator. The common denominator will then be \((x - 3)(x + 1)\).

Step 2: Convert each Fraction to Have the Common Denominator

Rewrite each fraction with the common denominator \((x - 3)(x + 1)\). So the expressions become: \[\frac{x^{2} + 9x}{(x - 3)(x + 1)} - \frac{5}{-(x - 3)(x + 1)}\]

Step 3: Combine the Fractions

As both fractions have the same denominator, they can be combined into one fraction: \[\frac{x^{2} + 9x + 5}{(x - 3)(x + 1)}\].

Step 4: Simplify, if Possible

There are no further simplifications possible for the fraction \(\frac{x^{2} + 9x + 5}{(x - 3)(x + 1)}\). Thus, this is the final result.