Question

Solve the equation by multiplying each side by the least common denominator. $$2+\frac{8}{x-5}=\frac{x+5}{x^{2}-25}$$

Short answer

The solution to the equation is \(x = 1\) and \(x = 7\). They are the only solutions that do not make the original equation undefined.

Step-by-step solution

Identify the least common denominator

First, identify the least common denominator (LCD). In this case, that is \(x^{2}-25\). This is because it's the only denominator that isn't already factored into the others.

Multiply each side by the least common denominator

Multiply each side by \(x^{2}-25\) to eliminate the fractions: \((2+(8/(x-5))) * (x^{2}-25)=(x+5)\). Simplify this to \(2x^{2}-50 + 8x - 40 = x^{3}+5x^{2}-25x-125\).

Simplify the equation and solve for x

Rearrange and simplify the equation to set it equal to zero: \(x^{3}+5x^{2}-25x-125 -2x^{2}+50 - 8x +40=0\). This simplifies to \(x^{3} + 3x^{2} - 33x - 35 = 0\). By solving this equation, you get three possible solutions for x: \(x = -5, x = 1, x = 7\). However, the solution \(x = -5\) should be discarded because it would make the original equation undefined.