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

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

Solve for x (complex solution)

$$x=\sqrt{3}-1\approx 0.732050808$$

$$x=-\left(\sqrt{3}+1\right)\approx -2.732050808$$

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

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

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

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

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

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

Combine -x^{2} and x^{2} to get 0.

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

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

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

Subtract x^{2} from both sides.

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

Add x to both sides.

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

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

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

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.

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

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

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

Square -2.

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

Multiply -4 times -1.

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

Multiply 4 times 2.

$$x=\frac{-\left(-2\right)±\sqrt{4+8}}{2\left(-1\right)}$$

Add 4 to 8.

$$x=\frac{-\left(-2\right)±\sqrt{12}}{2\left(-1\right)}$$

Take the square root of 12.

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

The opposite of -2 is 2.

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

Multiply 2 times -1.

$$x=\frac{2±2\sqrt{3}}{-2}$$

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

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

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

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

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

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

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

$$x=\sqrt{3}-1$$

The equation is now solved.

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

Steps for Completing the Square

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

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

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

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

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

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

Combine -x^{2} and x^{2} to get 0.

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

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

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

Subtract x^{2} from both sides.

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

Add x to both sides.

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

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

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

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

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

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.

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide -2 by -1.

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

Divide -2 by -1.

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

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.

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

Square 1.

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

Add 2 to 1.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+1=\sqrt{3} x+1=-\sqrt{3}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{3}-1 x=-\sqrt{3}-1$$

Solve for x

$$x=\sqrt{3}-1\approx 0.732050808$$

$$x=-\sqrt{3}-1\approx -2.732050808$$

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

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

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

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

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

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

Combine -x^{2} and x^{2} to get 0.

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

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

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

Subtract x^{2} from both sides.

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

Add x to both sides.

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

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

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

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.

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

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

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

Square -2.

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

Multiply -4 times -1.

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

Multiply 4 times 2.

$$x=\frac{-\left(-2\right)±\sqrt{4+8}}{2\left(-1\right)}$$

Add 4 to 8.

$$x=\frac{-\left(-2\right)±\sqrt{12}}{2\left(-1\right)}$$

Take the square root of 12.

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

The opposite of -2 is 2.

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

Multiply 2 times -1.

$$x=\frac{2±2\sqrt{3}}{-2}$$

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

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

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

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

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

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

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

$$x=\sqrt{3}-1$$

The equation is now solved.

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

Steps for Completing the Square

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

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

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

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

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

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

Combine -x^{2} and x^{2} to get 0.

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

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

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

Subtract x^{2} from both sides.

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

Add x to both sides.

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

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

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

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

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

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.

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide -2 by -1.

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

Divide -2 by -1.

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

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.

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

Square 1.

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

Add 2 to 1.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+1=\sqrt{3} x+1=-\sqrt{3}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{3}-1 x=-\sqrt{3}-1$$