Problem 52 - College Algebra Essentials

Question

Add or subtract as indicated. $$\frac{3}{5 x+2}+\frac{5 x}{25 x^{2}-4}$$

Step-by-Step Solution

Answer

\(\frac{20x - 6}{(5x+2)(5x-2)}\)

Step 1: Simplify the Denominator

The second fraction can be simplified by identifying its denominator as the difference of squares. Hence \(25x^{2}-4\) can be written as \((5x+2)(5x-2)\). Then, the fraction becomes \(\frac{5x}{(5x+2)(5x-2)}\)

Step 2: Identify the Common Denominator

Comparing both fractions, observe that the common denominator is \((5x+2)(5x-2)\). So, rewrite the first fraction \(\frac{3}{(5x+2)}\) as \(\frac{3(5x-2)}{(5x+2)(5x-2)}\) to match the denominators

Step 3: Add the Fractions

With a shared denominator in both fractions, the add operation becomes straightforward: \( \frac{3(5x-2)+ 5x}{(5x+2)(5x-2)}\)

Step 4: Simplify the result

Expand and simplify the numerator to obtain the final result: \( \frac{15x - 6 + 5x}{(5x+2)(5x-2)} = \frac{20x - 6}{(5x+2)(5x-2)}\)