# How do you solve 5/(n^3+5n^2)=4/(n+5)+1/n^2 and check for extraneous solutions?

Nov 4, 2016

Let's start by determining the LCD, or the Least Common Denominator.

$\frac{5}{{n}^{2} \left(n + 5\right)} = \frac{4}{n + 5} + \frac{1}{n} ^ 2$

$\frac{5}{{n}^{2} \left(n + 5\right)} = \frac{4 \left({n}^{2}\right)}{{n}^{2} \left(n + 5\right)} + \frac{n + 5}{{n}^{2} \left(n + 5\right)}$

We can now eliminate the denominators.

$5 = 4 {n}^{2} + n + 5$

$0 = 4 {n}^{2} + n$

$0 = n \left(4 n + 1\right)$

$n = 0 \mathmr{and} - \frac{1}{4}$

However, $n = 0$ is extraneous since it is one of the restrictions on the original equation.

Hence, $n = - \frac{1}{4}$ is the only actual solution.

Hopefully this helps!