Solve n=5/n+2

$$n = \frac { 5 } { n + 2 }$$

Solve for n

$$n=\sqrt{6}-1\approx 1.449489743$$

$$n=-\sqrt{6}-1\approx -3.449489743$$

Steps Using the Quadratic Formula

Subtract \frac{5}{n+2} from both sides.

$$n-\frac{5}{n+2}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply n times \frac{n+2}{n+2}.

$$\frac{n\left(n+2\right)}{n+2}-\frac{5}{n+2}=0$$

Since \frac{n\left(n+2\right)}{n+2} and \frac{5}{n+2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{n\left(n+2\right)-5}{n+2}=0$$

Do the multiplications in n\left(n+2\right)-5.

$$\frac{n^{2}+2n-5}{n+2}=0$$

Variable n cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by n+2.

$$n^{2}+2n-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-2±\sqrt{2^{2}-4\left(-5\right)}}{2}$$

Square 2.

$$n=\frac{-2±\sqrt{4-4\left(-5\right)}}{2}$$

Multiply -4 times -5.

$$n=\frac{-2±\sqrt{4+20}}{2}$$

Add 4 to 20.

$$n=\frac{-2±\sqrt{24}}{2}$$

Take the square root of 24.

$$n=\frac{-2±2\sqrt{6}}{2}$$

Now solve the equation n=\frac{-2±2\sqrt{6}}{2} when ± is plus. Add -2 to 2\sqrt{6}.

$$n=\frac{2\sqrt{6}-2}{2}$$

Divide -2+2\sqrt{6} by 2.

$$n=\sqrt{6}-1$$

Now solve the equation n=\frac{-2±2\sqrt{6}}{2} when ± is minus. Subtract 2\sqrt{6} from -2.

$$n=\frac{-2\sqrt{6}-2}{2}$$

Divide -2-2\sqrt{6} by 2.

$$n=-\sqrt{6}-1$$

The equation is now solved.

$$n=\sqrt{6}-1 n=-\sqrt{6}-1$$

Steps for Completing the Square

Subtract \frac{5}{n+2} from both sides.

$$n-\frac{5}{n+2}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply n times \frac{n+2}{n+2}.

$$\frac{n\left(n+2\right)}{n+2}-\frac{5}{n+2}=0$$

Since \frac{n\left(n+2\right)}{n+2} and \frac{5}{n+2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{n\left(n+2\right)-5}{n+2}=0$$

Do the multiplications in n\left(n+2\right)-5.

$$\frac{n^{2}+2n-5}{n+2}=0$$

Variable n cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by n+2.

$$n^{2}+2n-5=0$$

Add 5 to both sides. Anything plus zero gives itself.

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

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+2n+1^{2}=5+1^{2}$$

Square 1.

$$n^{2}+2n+1=5+1$$

Add 5 to 1.

$$n^{2}+2n+1=6$$

Factor n^{2}+2n+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n+1\right)^{2}=6$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+1\right)^{2}}=\sqrt{6}$$

Simplify.

$$n+1=\sqrt{6} n+1=-\sqrt{6}$$

Subtract 1 from both sides of the equation.

$$n=\sqrt{6}-1 n=-\sqrt{6}-1$$