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

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

Solve for x

$$x=4$$

$$x=0$$

Steps Using the Quadratic Formula

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

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

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

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

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

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

To find the opposite of x^{2}+x-2, find the opposite of each term.

$$x^{2}-3x+2-x^{2}-x+2=4-x^{2}$$

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

$$-3x+2-x+2=4-x^{2}$$

Combine -3x and -x to get -4x.

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

Add 2 and 2 to get 4.

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

Subtract 4 from both sides.

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

Subtract 4 from 4 to get 0.

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

Add x^{2} to both sides.

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

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.

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

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

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

Take the square root of \left(-4\right)^{2}.

$$x=\frac{-\left(-4\right)±4}{2}$$

The opposite of -4 is 4.

$$x=\frac{4±4}{2}$$

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

$$x=\frac{8}{2}$$

Divide 8 by 2.

$$x=4$$

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

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

Divide 0 by 2.

$$x=0$$

The equation is now solved.

$$x=4 x=0$$

Steps for Completing the Square

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

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

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

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

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

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

To find the opposite of x^{2}+x-2, find the opposite of each term.

$$x^{2}-3x+2-x^{2}-x+2=4-x^{2}$$

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

$$-3x+2-x+2=4-x^{2}$$

Combine -3x and -x to get -4x.

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

Add 2 and 2 to get 4.

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

Add x^{2} to both sides.

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

Subtract 4 from both sides.

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

Subtract 4 from 4 to get 0.

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

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.

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

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -2.

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

Factor x^{2}-4x+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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-2=2 x-2=-2$$

Add 2 to both sides of the equation.

$$x=4 x=0$$