Solve n^2/n+1=5

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

Solve for n

$$n = \frac{3 \sqrt{5} + 5}{2} \approx 5.854101966$$

$$n=\frac{5-3\sqrt{5}}{2}\approx -0.854101966$$

Steps Using the Quadratic Formula

Variable n cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by n+1.

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

Use the distributive property to multiply 5 by n+1.

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

Subtract 5n from both sides.

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

Subtract 5 from both sides.

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

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

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

Square -5.

$$n=\frac{-\left(-5\right)±\sqrt{25-4\left(-5\right)}}{2}$$

Multiply -4 times -5.

$$n=\frac{-\left(-5\right)±\sqrt{25+20}}{2}$$

Add 25 to 20.

$$n=\frac{-\left(-5\right)±\sqrt{45}}{2}$$

Take the square root of 45.

$$n=\frac{-\left(-5\right)±3\sqrt{5}}{2}$$

The opposite of -5 is 5.

$$n=\frac{5±3\sqrt{5}}{2}$$

Now solve the equation n=\frac{5±3\sqrt{5}}{2} when ± is plus. Add 5 to 3\sqrt{5}.

$$n=\frac{3\sqrt{5}+5}{2}$$

Now solve the equation n=\frac{5±3\sqrt{5}}{2} when ± is minus. Subtract 3\sqrt{5} from 5.

$$n=\frac{5-3\sqrt{5}}{2}$$

The equation is now solved.

$$n=\frac{3\sqrt{5}+5}{2} n=\frac{5-3\sqrt{5}}{2}$$

Steps for Completing the Square

Variable n cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by n+1.

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

Use the distributive property to multiply 5 by n+1.

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

Subtract 5n from both sides.

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

Divide -5, the coefficient of the x term, by 2 to get -\frac{5}{2}. Then add the square of -\frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-5n+\frac{25}{4}=5+\frac{25}{4}$$

Add 5 to \frac{25}{4}.

$$n^{2}-5n+\frac{25}{4}=\frac{45}{4}$$

Factor n^{2}-5n+\frac{25}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{5}{2}\right)^{2}=\frac{45}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{5}{2}\right)^{2}}=\sqrt{\frac{45}{4}}$$

Simplify.

$$n-\frac{5}{2}=\frac{3\sqrt{5}}{2} n-\frac{5}{2}=-\frac{3\sqrt{5}}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$n=\frac{3\sqrt{5}+5}{2} n=\frac{5-3\sqrt{5}}{2}$$