1Cademy - Subtracting $$\frac{2x}{x^2-4}

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How to Add or Subtract Rational Expressions

Example

Subtracting $\frac{2x}{x^2-4} - \frac{1}{x+2}$

Subtract two rational expressions where one denominator is a difference of squares that contains the other denominator as a factor:

$\frac{2x}{x^2 - 4} - \frac{1}{x + 2}$

Step 1 — Find the LCD and rewrite each fraction. Factor the first denominator as a difference of squares: $x^2 - 4 = (x - 2)(x + 2)$ . The second denominator $(x + 2)$ is already one of these factors. The LCD is $(x - 2)(x + 2)$ .

The first fraction already has the LCD. The second fraction is missing the factor $(x - 2)$ ; multiply its numerator and denominator by $(x - 2)$ :

$\frac{2x}{(x - 2)(x + 2)} - \frac{1(x - 2)}{(x - 2)(x + 2)}$

Distribute in the second numerator: $1(x - 2) = x - 2$ .

Step 2 — Subtract the rational expressions. Subtract the numerators over the common denominator, placing the second numerator in parentheses:

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

Distribute the negative sign:

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

Combine like terms:

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

Step 3 — Simplify, if possible. The numerator $(x + 2)$ shares a common factor with the denominator. Cancel the shared factor $(x + 2)$ :

$\frac{x + 2}{(x - 2)(x + 2)} = \frac{1}{x - 2}$

This example demonstrates the subtraction process where finding the LCD involves factoring a difference of squares. After converting to a common denominator and distributing the negative sign across the second numerator, the resulting combined numerator shares a factor with the denominator, which cancels out to produce the simplified result.

Updated 2026-05-25

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OpenStax Intermediate Algebra

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