Solve 1/s_{h}=1/2n^2+2n

$$\frac { 1 } { s _ { h } } = \frac { 1 } { 2 n ^ { 2 } + 2 n }$$

Solve for s_h

$$s_{h}=2n\left(n+1\right)$$

$$n\neq -1\text{ and }n\neq 0$$

Solution Steps

Variable s_{h} cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2ns_{h}\left(n+1\right), the least common multiple of s_{h},2n^{2}+2n.

$$2n\left(n+1\right)=s_{h}$$

Use the distributive property to multiply 2n by n+1.

$$2n^{2}+2n=s_{h}$$

Swap sides so that all variable terms are on the left hand side.

$$s_{h}=2n^{2}+2n$$

Variable s_{h} cannot be equal to 0.

$$s_{h}=2n^{2}+2n\text{, }s_{h}\neq 0$$

Solve for n

$$n=\frac{\sqrt{2s_{h}+1}-1}{2}$$

$$n=\frac{-\sqrt{2s_{h}+1}-1}{2}\text{, }s_{h}\neq 0\text{ and }s_{h}\geq -\frac{1}{2}$$