Solve x/1-x=1-x

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

Solve for x

$$x = \frac{\sqrt{5} + 3}{2} \approx 2.618033989$$

$$x=\frac{3-\sqrt{5}}{2}\approx 0.381966011$$

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.

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

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

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

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

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

Add 2x to both sides.

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

Combine x and 2x to get 3x.

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

Subtract 1 from both sides.

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

Subtract x^{2} from both sides.

$$3x-1-x^{2}=0$$

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.

$$-x^{2}+3x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 3 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-1\right)\left(-1\right)}}{2\left(-1\right)}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-1\right)\left(-1\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-3±\sqrt{9+4\left(-1\right)}}{2\left(-1\right)}$$

Multiply 4 times -1.

$$x=\frac{-3±\sqrt{9-4}}{2\left(-1\right)}$$

Add 9 to -4.

$$x=\frac{-3±\sqrt{5}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-3±\sqrt{5}}{-2}$$

Now solve the equation x=\frac{-3±\sqrt{5}}{-2} when ± is plus. Add -3 to \sqrt{5}.

$$x=\frac{\sqrt{5}-3}{-2}$$

Divide -3+\sqrt{5} by -2.

$$x=\frac{3-\sqrt{5}}{2}$$

Now solve the equation x=\frac{-3±\sqrt{5}}{-2} when ± is minus. Subtract \sqrt{5} from -3.

$$x=\frac{-\sqrt{5}-3}{-2}$$

Divide -3-\sqrt{5} by -2.

$$x=\frac{\sqrt{5}+3}{2}$$

The equation is now solved.

$$x=\frac{3-\sqrt{5}}{2} x=\frac{\sqrt{5}+3}{2}$$

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.

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

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

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

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

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

Add 2x to both sides.

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

Combine x and 2x to get 3x.

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

Subtract x^{2} from both sides.

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

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.

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide 3 by -1.

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

Divide 1 by -1.

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

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=-1+\frac{9}{4}$$

Add -1 to \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{5}{4}$$

Factor x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{5}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{5}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{\sqrt{5}}{2} x-\frac{3}{2}=-\frac{\sqrt{5}}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$x=\frac{\sqrt{5}+3}{2} x=\frac{3-\sqrt{5}}{2}$$