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

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

Solve for x (complex solution)

$$x=\frac{-\sqrt{3}i-1}{2}\approx -0.5-0.866025404i$$

$$x=\frac{-1+\sqrt{3}i}{2}\approx -0.5+0.866025404i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,-1,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+1\right)\left(x+2\right), the least common multiple of x-1,x+1,x+2.

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

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

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

Use the distributive property to multiply x^{2}+3x+2 by -1.

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

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

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

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

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

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

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

Subtract 2 from -2 to get -4.

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

Use the distributive property to multiply x^{2}-1 by 2.

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

Subtract 2x^{2} from both sides.

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

Add 2 to both sides.

$$-2x-4-2x^{2}+2=0$$

Add -4 and 2 to get -2.

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

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.

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

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

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

Square -2.

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

Multiply -4 times -2.

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

Multiply 8 times -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-16}}{2\left(-2\right)}$$

Add 4 to -16.

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

Take the square root of -12.

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

The opposite of -2 is 2.

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

Multiply 2 times -2.

$$x=\frac{2±2\sqrt{3}i}{-4}$$

Now solve the equation x=\frac{2±2\sqrt{3}i}{-4} when ± is plus. Add 2 to 2i\sqrt{3}.

$$x=\frac{2+2\sqrt{3}i}{-4}$$

Divide 2+2i\sqrt{3} by -4.

$$x=\frac{-\sqrt{3}i-1}{2}$$

Now solve the equation x=\frac{2±2\sqrt{3}i}{-4} when ± is minus. Subtract 2i\sqrt{3} from 2.

$$x=\frac{-2\sqrt{3}i+2}{-4}$$

Divide 2-2i\sqrt{3} by -4.

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

The equation is now solved.

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

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,-1,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+1\right)\left(x+2\right), the least common multiple of x-1,x+1,x+2.

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

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

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

Use the distributive property to multiply x^{2}+3x+2 by -1.

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

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

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

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

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

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

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

Subtract 2 from -2 to get -4.

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

Use the distributive property to multiply x^{2}-1 by 2.

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

Subtract 2x^{2} from both sides.

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

Add 4 to both sides.

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

Add -2 and 4 to get 2.

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

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.

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

Divide both sides by -2.

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

Dividing by -2 undoes the multiplication by -2.

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

Divide -2 by -2.

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

Divide 2 by -2.

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

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.

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

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

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

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

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Subtract \frac{1}{2} from both sides of the equation.

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