1Cademy - Simplifying $$\frac{\frac{2}{x+6}}{\frac{4}{x-6}-\frac{4}{x^2-36}}$$ Using the LCD

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Simplifying a Complex Rational Expression by Using the LCD

Example

Simplifying $\frac{\frac{2}{x+6}}{\frac{4}{x-6}-\frac{4}{x^2-36}}$ Using the LCD

Simplify the complex rational expression using the LCD method:

$\frac{\frac{2}{x+6}}{\frac{4}{x-6}-\frac{4}{x^2-36}}$

Step 1 — Find the LCD of all inner fractions. The inner denominators are $(x+6)$ , $(x-6)$ , and $x^2-36$ . Recognize $x^2-36$ as a difference of squares: $x^2-36 = (x+6)(x-6)$ . Since this already contains the other two denominators as factors, the LCD is $(x+6)(x-6)$ .

Step 2 — Multiply the numerator and denominator of the complex fraction by the LCD $(x+6)(x-6)$ . Distribute to each term:

$\frac{(x+6)(x-6) \cdot \frac{2}{x+6}}{(x+6)(x-6) \cdot \frac{4}{x-6} - (x+6)(x-6) \cdot \frac{4}{(x+6)(x-6)}}$

Step 3 — Simplify by canceling matching factors. In the numerator, $(x+6)$ cancels, leaving $2(x-6)$ . In the first denominator term, $(x-6)$ cancels, giving $4(x+6)$ . In the second denominator term, $(x+6)(x-6)$ cancels entirely, leaving $4$ :

$\frac{2(x-6)}{4(x+6) - 4}$

Step 4 — Simplify the denominator. Distribute and combine like terms: $4(x+6) - 4 = 4x + 24 - 4 = 4x + 20$ . Factor: $4x + 20 = 4(x+5)$ :

$\frac{2(x-6)}{4(x+5)}$

Step 5 — Remove common factors. The numerator and denominator share the factor $2$ . Cancel it:

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

No additional factors are shared between the numerator and denominator, so the expression is fully simplified. This example demonstrates how the LCD method handles a complex rational expression whose inner denominators involve both linear binomials and a difference of squares — factoring $x^2-36$ into $(x+6)(x-6)$ is the key step in identifying the LCD.

Updated 2026-04-30

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