Solve x+1/x+2-2x+1/x-2=0

$$\frac { x + 1 } { x + 2 } - \frac { 2 x + 1 } { x - 2 } = 0$$

Solve for x

$$x=\sqrt{5}-3\approx -0.763932023$$

$$x=-\left(\sqrt{5}+3\right)\approx -5.236067977$$

Steps Using the Quadratic Formula

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

$$\left(x-2\right)\left(x+1\right)-\left(x+2\right)\left(2x+1\right)=0$$

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

$$x^{2}-x-2-\left(x+2\right)\left(2x+1\right)=0$$

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

$$x^{2}-x-2-\left(2x^{2}+5x+2\right)=0$$

To find the opposite of 2x^{2}+5x+2, find the opposite of each term.

$$x^{2}-x-2-2x^{2}-5x-2=0$$

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

$$-x^{2}-x-2-5x-2=0$$

Combine -x and -5x to get -6x.

$$-x^{2}-6x-2-2=0$$

Subtract 2 from -2 to get -4.

$$-x^{2}-6x-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -6 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-1\right)\left(-4\right)}}{2\left(-1\right)}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-1\right)\left(-4\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-6\right)±\sqrt{36+4\left(-4\right)}}{2\left(-1\right)}$$

Multiply 4 times -4.

$$x=\frac{-\left(-6\right)±\sqrt{36-16}}{2\left(-1\right)}$$

Add 36 to -16.

$$x=\frac{-\left(-6\right)±\sqrt{20}}{2\left(-1\right)}$$

Take the square root of 20.

$$x=\frac{-\left(-6\right)±2\sqrt{5}}{2\left(-1\right)}$$

The opposite of -6 is 6.

$$x=\frac{6±2\sqrt{5}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{6±2\sqrt{5}}{-2}$$

Now solve the equation x=\frac{6±2\sqrt{5}}{-2} when ± is plus. Add 6 to 2\sqrt{5}.

$$x=\frac{2\sqrt{5}+6}{-2}$$

Divide 6+2\sqrt{5} by -2.

$$x=-\left(\sqrt{5}+3\right)$$

Now solve the equation x=\frac{6±2\sqrt{5}}{-2} when ± is minus. Subtract 2\sqrt{5} from 6.

$$x=\frac{6-2\sqrt{5}}{-2}$$

Divide 6-2\sqrt{5} by -2.

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

The equation is now solved.

$$x=-\left(\sqrt{5}+3\right) x=\sqrt{5}-3$$

Steps for Completing the Square

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

$$\left(x-2\right)\left(x+1\right)-\left(x+2\right)\left(2x+1\right)=0$$

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

$$x^{2}-x-2-\left(x+2\right)\left(2x+1\right)=0$$

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

$$x^{2}-x-2-\left(2x^{2}+5x+2\right)=0$$

To find the opposite of 2x^{2}+5x+2, find the opposite of each term.

$$x^{2}-x-2-2x^{2}-5x-2=0$$

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

$$-x^{2}-x-2-5x-2=0$$

Combine -x and -5x to get -6x.

$$-x^{2}-6x-2-2=0$$

Subtract 2 from -2 to get -4.

$$-x^{2}-6x-4=0$$

Add 4 to both sides. Anything plus zero gives itself.

$$-x^{2}-6x=4$$

Divide both sides by -1.

$$\frac{-x^{2}-6x}{-1}=\frac{4}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{6}{-1}\right)x=\frac{4}{-1}$$

Divide -6 by -1.

$$x^{2}+6x=\frac{4}{-1}$$

Divide 4 by -1.

$$x^{2}+6x=-4$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+6x+3^{2}=-4+3^{2}$$

Square 3.

$$x^{2}+6x+9=-4+9$$

Add -4 to 9.

$$x^{2}+6x+9=5$$

Factor x^{2}+6x+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}.

$$\left(x+3\right)^{2}=5$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{5}$$

Simplify.

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

Subtract 3 from both sides of the equation.

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