Problem 52 - College Algebra

Question

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

Step-by-Step Solution

Answer

The result after adding the given fractions is \(\frac{20x - 6}{25x^{2}-4}\).

Step 1: Factoring the denominators

Factor the denominators to reveal their underlying structure. The first denominator is simple, it's \(5x + 2\), but the second one, \(25x^{2} - 4\), is a difference of squares, which can be factored as \((5x-2)(5x+2)\).

Step 2: Identifying the least common denominator (LCD)

Now identify the LCD, as this is necessary to add or subtract fractions. Here, the LCD is the product of both factored forms, which is \((5x+2)(5x-2)\). This is due to the factor \(5x+2\) being present in the first denominator and both \(5x+2\) and \(5x-2\) being present in the second denominator.

Step 3: Transforming the fractions into equivalent forms

To add or subtract fractions, they should have the same denominator. So, rewrite each fraction over the LCD: \((5x+2)(5x-2)\). The first fraction becomes: \[\frac{3(5x-2)}{(5x+2)(5x-2)}\]. The second fraction already has the LCD as the denominator, so it remains the same.

Step 4: Adding the fractions

Now that both fractions have the same denominator, they can be added: \[\frac{3(5x-2) + 5x}{(5x+2)(5x-2)}\]. This simplifies to: \[\frac{15x - 6 + 5x}{25x^{2}-4}.\]

Step 5: Simplifying the result

The final step is to simplify the result. The numerator can be simplified by combining the like terms, giving: \[\frac{20x - 6}{25x^{2}-4}.\] This is the final answer.