Solve x/x-1-1=x/2

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

Solve for x

$$x=-1$$

$$x=2$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-1\right), the least common multiple of x-1,2.

$$2x+2\left(x-1\right)\left(-1\right)=\left(x-1\right)x$$

Multiply 2 and -1 to get -2.

$$2x-2\left(x-1\right)=\left(x-1\right)x$$

Use the distributive property to multiply -2 by x-1.

$$2x-2x+2=\left(x-1\right)x$$

Combine 2x and -2x to get 0.

$$2=\left(x-1\right)x$$

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

$$2=x^{2}-x$$

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

$$x^{2}-x=2$$

Subtract 2 from both sides.

$$x^{2}-x-2=0$$

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

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

Multiply -4 times -2.

$$x=\frac{-\left(-1\right)±\sqrt{1+8}}{2}$$

Add 1 to 8.

$$x=\frac{-\left(-1\right)±\sqrt{9}}{2}$$

Take the square root of 9.

$$x=\frac{-\left(-1\right)±3}{2}$$

The opposite of -1 is 1.

$$x=\frac{1±3}{2}$$

Now solve the equation x=\frac{1±3}{2} when ± is plus. Add 1 to 3.

$$x=\frac{4}{2}$$

Divide 4 by 2.

$$x=2$$

Now solve the equation x=\frac{1±3}{2} when ± is minus. Subtract 3 from 1.

$$x=-\frac{2}{2}$$

Divide -2 by 2.

$$x=-1$$

The equation is now solved.

$$x=2 x=-1$$

Steps for Completing the Square

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-1\right), the least common multiple of x-1,2.

$$2x+2\left(x-1\right)\left(-1\right)=\left(x-1\right)x$$

Multiply 2 and -1 to get -2.

$$2x-2\left(x-1\right)=\left(x-1\right)x$$

Use the distributive property to multiply -2 by x-1.

$$2x-2x+2=\left(x-1\right)x$$

Combine 2x and -2x to get 0.

$$2=\left(x-1\right)x$$

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

$$2=x^{2}-x$$

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

$$x^{2}-x=2$$

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

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=2+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=2+\frac{1}{4}$$

Add 2 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{9}{4}$$

Factor x^{2}-x+\frac{1}{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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{3}{2} x-\frac{1}{2}=-\frac{3}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$x=2 x=-1$$