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

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

Solve for x

$$x=-4$$

$$x=0$$

Steps Using Factoring

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

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

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

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

Use the distributive property to multiply 2x+2 by x.

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

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

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

Subtract x^{2} from both sides.

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

Combine 2x^{2} and -x^{2} to get x^{2}.

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

Add 2x to both sides.

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

Combine 2x and 2x to get 4x.

$$x^{2}+4x=0$$

Factor out x.

$$x\left(x+4\right)=0$$

To find equation solutions, solve x=0 and x+4=0.

$$x=0 x=-4$$

Steps Using the Quadratic Formula

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

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

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

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

Use the distributive property to multiply 2x+2 by x.

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

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

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

Subtract x^{2} from both sides.

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

Combine 2x^{2} and -x^{2} to get x^{2}.

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

Add 2x to both sides.

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

Combine 2x and 2x to get 4x.

$$x^{2}+4x=0$$

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

$$x=\frac{-4±\sqrt{4^{2}}}{2}$$

Take the square root of 4^{2}.

$$x=\frac{-4±4}{2}$$

Now solve the equation x=\frac{-4±4}{2} when ± is plus. Add -4 to 4.

$$x=\frac{0}{2}$$

Divide 0 by 2.

$$x=0$$

Now solve the equation x=\frac{-4±4}{2} when ± is minus. Subtract 4 from -4.

$$x=-\frac{8}{2}$$

Divide -8 by 2.

$$x=-4$$

The equation is now solved.

$$x=0 x=-4$$

Steps for Completing the Square

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

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

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

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

Use the distributive property to multiply 2x+2 by x.

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

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

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

Subtract x^{2} from both sides.

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

Combine 2x^{2} and -x^{2} to get x^{2}.

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

Add 2x to both sides.

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

Combine 2x and 2x to get 4x.

$$x^{2}+4x=0$$

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+4x+2^{2}=2^{2}$$

Square 2.

$$x^{2}+4x+4=4$$

Factor x^{2}+4x+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}.

$$\left(x+2\right)^{2}=4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{4}$$

Simplify.

$$x+2=2 x+2=-2$$

Subtract 2 from both sides of the equation.

$$x=0 x=-4$$