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

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

Solve for x (complex solution)

$$x=\frac{1+\sqrt{7}i}{2}\approx 0.5+1.322875656i$$

$$x=\frac{-\sqrt{7}i+1}{2}\approx 0.5-1.322875656i$$

Steps Using the Quadratic Formula

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

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

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

$$2x=3x-x^{2}-2$$

Subtract 3x from both sides.

$$2x-3x=-x^{2}-2$$

Combine 2x and -3x to get -x.

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

Add x^{2} to both sides.

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

Add 2 to both sides.

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

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

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

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}.

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

Multiply -4 times 2.

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

Add 1 to -8.

$$x=\frac{-\left(-1\right)±\sqrt{-7}}{2}$$

Take the square root of -7.

$$x=\frac{-\left(-1\right)±\sqrt{7}i}{2}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{7}i}{2}$$

Now solve the equation x=\frac{1±\sqrt{7}i}{2} when ± is plus. Add 1 to i\sqrt{7}.

$$x=\frac{1+\sqrt{7}i}{2}$$

Now solve the equation x=\frac{1±\sqrt{7}i}{2} when ± is minus. Subtract i\sqrt{7} from 1.

$$x=\frac{-\sqrt{7}i+1}{2}$$

The equation is now solved.

$$x=\frac{1+\sqrt{7}i}{2} x=\frac{-\sqrt{7}i+1}{2}$$

Steps for Completing the Square

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

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

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

$$2x=3x-x^{2}-2$$

Subtract 3x from both sides.

$$2x-3x=-x^{2}-2$$

Combine 2x and -3x to get -x.

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

Add x^{2} to both sides.

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

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

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

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}.

$$x^{2}-x+\frac{1}{4}=-\frac{7}{4}$$

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}.

$$\left(x-\frac{1}{2}\right)^{2}=-\frac{7}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{-\frac{7}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{7}i}{2} x-\frac{1}{2}=-\frac{\sqrt{7}i}{2}$$

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

$$x=\frac{1+\sqrt{7}i}{2} x=\frac{-\sqrt{7}i+1}{2}$$