Problem 47 - College Algebra Essentials

Question

Add or subtract as indicated. $$\frac{x+5}{x-5}+\frac{x-5}{x+5}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{2(x^2 + 25)}{x^2 - 25}\)

Step 1: Identify a common denominator

The denominators of the fractions are \(x - 5\) and \(x + 5\). The least common denominator (LCD) of these two is found by simply multiplying them together as they have no common factor. So, the LCD is \((x - 5)(x + 5)\).

Step 2: Rewrite fractions with common denominator

Rewrite the fractions such that they both have the new common denominator. To do this, multiply both the numerator and denominator of each fraction by what they were lacking from the LCD. This gives us: \[\frac{(x + 5) (x + 5)}{(x - 5)(x + 5)} + \frac{(x - 5)(x - 5)}{(x - 5)(x + 5)}\]

Step 3: Simplify each fraction

Multiply out the numerators in both new fractions while leaving the denominators in factored form to give \[\frac{x^2 + 10x + 25}{(x - 5)(x + 5)} + \frac{x^2 - 10x + 25}{(x - 5)(x + 5)}\]

Step 4: Combine the fractions

Since both fractions have the same denominator, add the numerators which results in \[(x^2 + 10x + 25 + x^2 - 10x + 25)/[(x - 5)(x + 5)]\]

Step 5: Simplify the result

Combine like terms in the numerator to simplify \[(2x^2 + 50)/[(x - 5)(x + 5)]\]

Step 6: Rewrite the denominator

Expand the denominator, this gives \[2x^2 + 50]/[x^2 - 25]\], simplify to give: \[\frac{2(x^2 + 25)}{x^2 - 25}\]