Solve n-1/n=-2

$$n - \frac { 1 } { n } = - 2$$

Solve for n (complex solution)

$$n=\sqrt{2}-1\approx 0.414213562$$

$$n=-\left(\sqrt{2}+1\right)\approx -2.414213562$$

Steps Using the Quadratic Formula

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by n.

$$nn-1=-2n$$

Multiply n and n to get n^{2}.

$$n^{2}-1=-2n$$

Add 2n to both sides.

$$n^{2}-1+2n=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$n^{2}+2n-1=0$$

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

$$n=\frac{-2±\sqrt{2^{2}-4\left(-1\right)}}{2}$$

Square 2.

$$n=\frac{-2±\sqrt{4-4\left(-1\right)}}{2}$$

Multiply -4 times -1.

$$n=\frac{-2±\sqrt{4+4}}{2}$$

Add 4 to 4.

$$n=\frac{-2±\sqrt{8}}{2}$$

Take the square root of 8.

$$n=\frac{-2±2\sqrt{2}}{2}$$

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

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

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

$$n=\sqrt{2}-1$$

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

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

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

$$n=-\sqrt{2}-1$$

The equation is now solved.

$$n=\sqrt{2}-1 n=-\sqrt{2}-1$$

Steps for Completing the Square

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by n.

$$nn-1=-2n$$

Multiply n and n to get n^{2}.

$$n^{2}-1=-2n$$

Add 2n to both sides.

$$n^{2}-1+2n=0$$

Add 1 to both sides. Anything plus zero gives itself.

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

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.

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

Square 1.

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

Add 1 to 1.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$n+1=\sqrt{2} n+1=-\sqrt{2}$$

Subtract 1 from both sides of the equation.

$$n=\sqrt{2}-1 n=-\sqrt{2}-1$$

Solve for n

$$n=\sqrt{2}-1\approx 0.414213562$$

$$n=-\sqrt{2}-1\approx -2.414213562$$

Steps Using the Quadratic Formula

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by n.

$$nn-1=-2n$$

Multiply n and n to get n^{2}.

$$n^{2}-1=-2n$$

Add 2n to both sides.

$$n^{2}-1+2n=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$n^{2}+2n-1=0$$

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

$$n=\frac{-2±\sqrt{2^{2}-4\left(-1\right)}}{2}$$

Square 2.

$$n=\frac{-2±\sqrt{4-4\left(-1\right)}}{2}$$

Multiply -4 times -1.

$$n=\frac{-2±\sqrt{4+4}}{2}$$

Add 4 to 4.

$$n=\frac{-2±\sqrt{8}}{2}$$

Take the square root of 8.

$$n=\frac{-2±2\sqrt{2}}{2}$$

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

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

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

$$n=\sqrt{2}-1$$

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

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

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

$$n=-\sqrt{2}-1$$

The equation is now solved.

$$n=\sqrt{2}-1 n=-\sqrt{2}-1$$

Steps for Completing the Square

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by n.

$$nn-1=-2n$$

Multiply n and n to get n^{2}.

$$n^{2}-1=-2n$$

Add 2n to both sides.

$$n^{2}-1+2n=0$$

Add 1 to both sides. Anything plus zero gives itself.

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

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.

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

Square 1.

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

Add 1 to 1.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$n+1=\sqrt{2} n+1=-\sqrt{2}$$

Subtract 1 from both sides of the equation.

$$n=\sqrt{2}-1 n=-\sqrt{2}-1$$