Solve 2/x(x-1)=1+2/x-1

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

Solve for x

$$x=-2$$

Steps Using the Quadratic Formula

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

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

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

$$2=x^{2}-x+x\times 2$$

Combine -x and x\times 2 to get 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}.

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

Square 1.

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

Multiply -4 times -2.

$$x=\frac{-1±\sqrt{1+8}}{2}$$

Add 1 to 8.

$$x=\frac{-1±\sqrt{9}}{2}$$

Take the square root of 9.

$$x=\frac{-1±3}{2}$$

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

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

Divide 2 by 2.

$$x=1$$

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

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

Divide -4 by 2.

$$x=-2$$

The equation is now solved.

$$x=1 x=-2$$

Variable x cannot be equal to 1.

$$x=-2$$

Steps for Completing the Square

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

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

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

$$2=x^{2}-x+x\times 2$$

Combine -x and x\times 2 to get 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.

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

Subtract \frac{1}{2} from both sides of the equation.

$$x=1 x=-2$$

Variable x cannot be equal to 1.

$$x=-2$$