Solve 1+1/n-1=1/n^2-n

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

Solve for n

$$n=-1$$

Steps Using Difference of Squares

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

$$n\left(n-1\right)+n=1$$

Use the distributive property to multiply n by n-1.

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

Combine -n and n to get 0.

$$n^{2}=1$$

Subtract 1 from both sides.

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

Consider n^{2}-1. Rewrite n^{2}-1 as n^{2}-1^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

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

To find equation solutions, solve n-1=0 and n+1=0.

$$n=1 n=-1$$

Variable n cannot be equal to 1.

$$n=-1$$

Steps by Finding Square Root

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

$$n\left(n-1\right)+n=1$$

Use the distributive property to multiply n by n-1.

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

Combine -n and n to get 0.

$$n^{2}=1$$

Take the square root of both sides of the equation.

$$n=1 n=-1$$

Variable n cannot be equal to 1.

$$n=-1$$

Steps Using the Quadratic Formula

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

$$n\left(n-1\right)+n=1$$

Use the distributive property to multiply n by n-1.

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

Combine -n and n to get 0.

$$n^{2}=1$$

Subtract 1 from both sides.

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

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

$$n=\frac{0±\sqrt{0^{2}-4\left(-1\right)}}{2}$$

Square 0.

$$n=\frac{0±\sqrt{-4\left(-1\right)}}{2}$$

Multiply -4 times -1.

$$n=\frac{0±\sqrt{4}}{2}$$

Take the square root of 4.

$$n=\frac{0±2}{2}$$

Now solve the equation n=\frac{0±2}{2} when ± is plus. Divide 2 by 2.

$$n=1$$

Now solve the equation n=\frac{0±2}{2} when ± is minus. Divide -2 by 2.

$$n=-1$$

The equation is now solved.

$$n=1 n=-1$$

Variable n cannot be equal to 1.

$$n=-1$$