1Cademy - Solving $$\frac{2}{p+2} + \frac{4}{p-2} = \frac{p-1}{p^2-4}$$

Learn Before

Strategy to Solve Equations with Rational Expressions

Example

Solving $\frac{2}{p+2} + \frac{4}{p-2} = \frac{p-1}{p^2-4}$

Solve the rational equation $\frac{2}{p+2} + \frac{4}{p-2} = \frac{p-1}{p^2-4}$ by applying the five-step strategy for equations with rational expressions. This example features a quadratic denominator that factors as a difference of squares, making it the product of the other two denominators.

Step 1 — Identify restricted values. Factor the quadratic denominator: $p^2 - 4 = (p+2)(p-2)$ . Setting each linear factor equal to zero gives $p = -2$ and $p = 2$ . Record $p \neq -2$ and $p \neq 2$ .

Step 2 — Find the LCD. The three denominators are $(p+2)$ , $(p-2)$ , and $(p+2)(p-2)$ . Because the quadratic denominator already contains the other two as factors, the LCD is $(p+2)(p-2)$ .

Step 3 — Clear the fractions. Multiply both sides by the LCD $(p+2)(p-2)$ and distribute to each term. Cancel matching denominator factors: the first term becomes $2(p-2)$ , the second becomes $4(p+2)$ , and the right side becomes $p - 1$ :

$2(p-2) + 4(p+2) = p - 1$

Step 4 — Solve the resulting equation. Distribute: $2p - 4 + 4p + 8 = p - 1$ . Combine like terms: $6p + 4 = p - 1$ . Subtract $p$ from both sides: $5p + 4 = -1$ . Subtract $4$ : $5p = -5$ . Divide by $5$ :

$p = -1$

Step 5 — Check. The value $p = -1$ does not equal either restricted value ( $2$ or $-2$ ), so it is not extraneous. Substitute into the original equation:

$\frac{2}{-1+2} + \frac{4}{-1-2} = \frac{-1-1}{(-1)^2-4}$

$\frac{2}{1} + \frac{4}{-3} = \frac{-2}{-3}$

$2 - \frac{4}{3} = \frac{2}{3}$

$\frac{6}{3} - \frac{4}{3} = \frac{2}{3}$ ✓

The solution is $p = -1$ . This example demonstrates a rational equation in which one denominator is the difference of squares $p^2 - 4 = (p+2)(p-2)$ . Recognizing this factorization is the key step, because it reveals that the LCD is simply the quadratic denominator itself. After clearing fractions, the equation reduces to a linear equation with a single solution — unlike equations where clearing fractions produces a quadratic.

Updated 2026-04-21

Contributors are:

Who are from:

References

OpenStax Elementary Algebra

Learn Before

Related

Learn After