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

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

Solve for x

$$x=\frac{1}{2}=0.5$$

$$x=-\frac{1}{2}=-0.5$$

Steps Using Difference of Squares

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 2x\left(x-1\right), the least common multiple of 2x,2x-2,1-x.

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

Use the distributive property to multiply x-1 by 2x+1 and combine like terms.

$$2x^{2}-x-1+x=-2xx$$

Combine -x and x to get 0.

$$2x^{2}-1=-2xx$$

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

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

Add 2x^{2} to both sides.

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

Combine 2x^{2} and 2x^{2} to get 4x^{2}.

$$4x^{2}-1=0$$

Consider 4x^{2}-1. Rewrite 4x^{2}-1 as \left(2x\right)^{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).

$$\left(2x-1\right)\left(2x+1\right)=0$$

To find equation solutions, solve 2x-1=0 and 2x+1=0.

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

Steps by Finding Square Root

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 2x\left(x-1\right), the least common multiple of 2x,2x-2,1-x.

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

Use the distributive property to multiply x-1 by 2x+1 and combine like terms.

$$2x^{2}-x-1+x=-2xx$$

Combine -x and x to get 0.

$$2x^{2}-1=-2xx$$

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

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

Add 2x^{2} to both sides.

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

Combine 2x^{2} and 2x^{2} to get 4x^{2}.

$$4x^{2}-1=0$$

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

$$4x^{2}=1$$

Divide both sides by 4.

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

Take the square root of both sides of the equation.

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

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 2x\left(x-1\right), the least common multiple of 2x,2x-2,1-x.

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

Use the distributive property to multiply x-1 by 2x+1 and combine like terms.

$$2x^{2}-x-1+x=-2xx$$

Combine -x and x to get 0.

$$2x^{2}-1=-2xx$$

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

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

Add 2x^{2} to both sides.

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

Combine 2x^{2} and 2x^{2} to get 4x^{2}.

$$4x^{2}-1=0$$

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

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

Square 0.

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

Multiply -4 times 4.

$$x=\frac{0±\sqrt{-16\left(-1\right)}}{2\times 4}$$

Multiply -16 times -1.

$$x=\frac{0±\sqrt{16}}{2\times 4}$$

Take the square root of 16.

$$x=\frac{0±4}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{0±4}{8}$$

Now solve the equation x=\frac{0±4}{8} when ± is plus. Reduce the fraction \frac{4}{8} to lowest terms by extracting and canceling out 4.

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

Now solve the equation x=\frac{0±4}{8} when ± is minus. Reduce the fraction \frac{-4}{8} to lowest terms by extracting and canceling out 4.

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

The equation is now solved.

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