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

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

Solve for x

$$x = \frac{\sqrt{7} + 1}{2} \approx 1.822875656$$

$$x=\frac{1-\sqrt{7}}{2}\approx -0.822875656$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by -x+2.

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

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

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

Combine 4x and -x to get 3x.

$$x-1=-2x^{2}+3x+2$$

Add 2x^{2} to both sides.

$$x-1+2x^{2}=3x+2$$

Subtract 3x from both sides.

$$x-1+2x^{2}-3x=2$$

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

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

Subtract 2 from both sides.

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

Subtract 2 from -1 to get -3.

$$-2x-3+2x^{2}=0$$

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.

$$2x^{2}-2x-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -2 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 2\left(-3\right)}}{2\times 2}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 2\left(-3\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-2\right)±\sqrt{4-8\left(-3\right)}}{2\times 2}$$

Multiply -8 times -3.

$$x=\frac{-\left(-2\right)±\sqrt{4+24}}{2\times 2}$$

Add 4 to 24.

$$x=\frac{-\left(-2\right)±\sqrt{28}}{2\times 2}$$

Take the square root of 28.

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

The opposite of -2 is 2.

$$x=\frac{2±2\sqrt{7}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{2±2\sqrt{7}}{4}$$

Now solve the equation x=\frac{2±2\sqrt{7}}{4} when ± is plus. Add 2 to 2\sqrt{7}.

$$x=\frac{2\sqrt{7}+2}{4}$$

Divide 2+2\sqrt{7} by 4.

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

Now solve the equation x=\frac{2±2\sqrt{7}}{4} when ± is minus. Subtract 2\sqrt{7} from 2.

$$x=\frac{2-2\sqrt{7}}{4}$$

Divide 2-2\sqrt{7} by 4.

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

The equation is now solved.

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

Steps for Completing the Square

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by -x+2.

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

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

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

Combine 4x and -x to get 3x.

$$x-1=-2x^{2}+3x+2$$

Add 2x^{2} to both sides.

$$x-1+2x^{2}=3x+2$$

Subtract 3x from both sides.

$$x-1+2x^{2}-3x=2$$

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

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

Add 1 to both sides.

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

Add 2 and 1 to get 3.

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

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.

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

Divide both sides by 2.

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

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{2}{2}\right)x=\frac{3}{2}$$

Divide -2 by 2.

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

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.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=\frac{3}{2}+\left(-\frac{1}{2}\right)^{2}$$

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

$$x^{2}-x+\frac{1}{4}=\frac{3}{2}+\frac{1}{4}$$

Add \frac{3}{2} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

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

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.

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

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

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