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

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

Solve for x

$$x=2$$

$$x=0$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by -x+1.

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

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

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

Combine x and 2x to get 3x.

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

To find the opposite of -x+1, find the opposite of each term.

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

Subtract x from both sides.

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

Combine 3x and -x to get 2x.

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

Add 1 to both sides.

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

Add -1 and 1 to get 0.

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

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

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

Take the square root of 2^{2}.

$$x=\frac{-2±2}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-2±2}{-2}$$

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

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

Divide 0 by -2.

$$x=0$$

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

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

Divide -4 by -2.

$$x=2$$

The equation is now solved.

$$x=0 x=2$$

Steps for Completing the Square

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by -x+1.

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

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

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

Combine x and 2x to get 3x.

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

To find the opposite of -x+1, find the opposite of each term.

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

Subtract x from both sides.

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

Combine 3x and -x to get 2x.

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

Add 1 to both sides.

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

Add -1 and 1 to get 0.

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide 2 by -1.

$$x^{2}-2x=\frac{0}{-1}$$

Divide 0 by -1.

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

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-1=1 x-1=-1$$

Add 1 to both sides of the equation.

$$x=2 x=0$$