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

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

Solve for x

$$x = \frac{2 \sqrt{10} + 2}{3} \approx 2.774851773$$

$$x=\frac{2-2\sqrt{10}}{3}\approx -1.44151844$$

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,x+2,x+1.

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

Use the distributive property to multiply x+2 by 2.

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

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

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

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

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

To find the opposite of 2x^{2}-2x-4, find the opposite of each term.

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

Combine 2x and 2x to get 4x.

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

Add 4 and 4 to get 8.

$$4x+8-2x^{2}=x^{2}-4$$

Subtract x^{2} from both sides.

$$4x+8-2x^{2}-x^{2}=-4$$

Combine -2x^{2} and -x^{2} to get -3x^{2}.

$$4x+8-3x^{2}=-4$$

Add 4 to both sides.

$$4x+8-3x^{2}+4=0$$

Add 8 and 4 to get 12.

$$4x+12-3x^{2}=0$$

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.

$$-3x^{2}+4x+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 4 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-3\right)\times 12}}{2\left(-3\right)}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-3\right)\times 12}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-4±\sqrt{16+12\times 12}}{2\left(-3\right)}$$

Multiply 12 times 12.

$$x=\frac{-4±\sqrt{16+144}}{2\left(-3\right)}$$

Add 16 to 144.

$$x=\frac{-4±\sqrt{160}}{2\left(-3\right)}$$

Take the square root of 160.

$$x=\frac{-4±4\sqrt{10}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-4±4\sqrt{10}}{-6}$$

Now solve the equation x=\frac{-4±4\sqrt{10}}{-6} when ± is plus. Add -4 to 4\sqrt{10}.

$$x=\frac{4\sqrt{10}-4}{-6}$$

Divide -4+4\sqrt{10} by -6.

$$x=\frac{2-2\sqrt{10}}{3}$$

Now solve the equation x=\frac{-4±4\sqrt{10}}{-6} when ± is minus. Subtract 4\sqrt{10} from -4.

$$x=\frac{-4\sqrt{10}-4}{-6}$$

Divide -4-4\sqrt{10} by -6.

$$x=\frac{2\sqrt{10}+2}{3}$$

The equation is now solved.

$$x=\frac{2-2\sqrt{10}}{3} x=\frac{2\sqrt{10}+2}{3}$$

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,x+2,x+1.

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

Use the distributive property to multiply x+2 by 2.

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

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

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

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

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

To find the opposite of 2x^{2}-2x-4, find the opposite of each term.

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

Combine 2x and 2x to get 4x.

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

Add 4 and 4 to get 8.

$$4x+8-2x^{2}=x^{2}-4$$

Subtract x^{2} from both sides.

$$4x+8-2x^{2}-x^{2}=-4$$

Combine -2x^{2} and -x^{2} to get -3x^{2}.

$$4x+8-3x^{2}=-4$$

Subtract 8 from both sides.

$$4x-3x^{2}=-4-8$$

Subtract 8 from -4 to get -12.

$$4x-3x^{2}=-12$$

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.

$$-3x^{2}+4x=-12$$

Divide both sides by -3.

$$\frac{-3x^{2}+4x}{-3}=-\frac{12}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{4}{-3}x=-\frac{12}{-3}$$

Divide 4 by -3.

$$x^{2}-\frac{4}{3}x=-\frac{12}{-3}$$

Divide -12 by -3.

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

Divide -\frac{4}{3}, the coefficient of the x term, by 2 to get -\frac{2}{3}. Then add the square of -\frac{2}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

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

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

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

$$x^{2}-\frac{4}{3}x+\frac{4}{9}=\frac{40}{9}$$

Factor x^{2}-\frac{4}{3}x+\frac{4}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{2}{3}\right)^{2}=\frac{40}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{2}{3}\right)^{2}}=\sqrt{\frac{40}{9}}$$

Simplify.

$$x-\frac{2}{3}=\frac{2\sqrt{10}}{3} x-\frac{2}{3}=-\frac{2\sqrt{10}}{3}$$

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

$$x=\frac{2\sqrt{10}+2}{3} x=\frac{2-2\sqrt{10}}{3}$$