How do you solve #x/(x+2) - 2/(x-2) = (x^2+4)/(x^2-4)#?

Question

Avery Alexander · Accepted Answer

Multiply through by #(x-2)(x+2)# and simplify to attempt to find a solution, but the only possibility turns out to be a spurious solution.

Given: #x/(x+2)-2/(x-2) = (x^2+4)/(x^2-4)# Note that #x^2-4 = (x-2)(x+2)# Multiply both sides by #(x-2)(x+2)# to get: #x(x-2)-2(x+2) = x^2+4# Note that this potentially (and does) introduces spurious solutions for #x = +-2# The left side can be made simpler like this: #x(x-2)-2(x+2) = x^2-2x-2x-4 = x^2-4x-4# Thus, the formula becomes: #color(red)(cancel(color(black)(x^2)))-4x-4 = color(red)(cancel(color(black)(x^2)))+4# Subtract #x^2# from both sides and add #4# to both sides to get: #-4x = 8# Divide both sides by #-4# to get: #x=-2# This is not a solution of the original equation, since if #x=-2# then the denominators of two of the expressions are zero. Thus, there are no answers for the initial issue.

Jace Anderson · Answer

undefined To solve the equation x/(x+2) - 2/(x-2) = (x^2+4)/(x^2-4), we can follow these steps:

1. Start by finding a common denominator for all the fractions involved. In this case, the common denominator is (x+2)(x-2).

2. Multiply each term by the common denominator to eliminate the fractions.

3. Simplify the equation by distributing and combining like terms.

4. Continue simplifying until you have a quadratic equation.

5. Solve the quadratic equation by factoring, using the quadratic formula, or completing the square.

6. Check your solutions by substituting them back into the original equation to ensure they satisfy the equation.

By following these steps, you will be able to solve the given equation.