Solve 3/x+2=1/x-2+1/x+1

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

Solve for x

$$x=\sqrt{13}+3\approx 6.605551275$$

$$x=3-\sqrt{13}\approx -0.605551275$$

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+1.

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

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

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

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

$$3x^{2}-3x-6=\left(x+1\right)\left(x+2\right)+x^{2}-4$$

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

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

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

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

Subtract 4 from 2 to get -2.

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

Subtract 2x^{2} from both sides.

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

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

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

Subtract 3x from both sides.

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

Combine -3x and -3x to get -6x.

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

Add 2 to both sides.

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

Add -6 and 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}.

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

Square -6.

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

Multiply -4 times -4.

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

Add 36 to 16.

$$x=\frac{-\left(-6\right)±\sqrt{52}}{2}$$

Take the square root of 52.

$$x=\frac{-\left(-6\right)±2\sqrt{13}}{2}$$

The opposite of -6 is 6.

$$x=\frac{6±2\sqrt{13}}{2}$$

Now solve the equation x=\frac{6±2\sqrt{13}}{2} when ± is plus. Add 6 to 2\sqrt{13}.

$$x=\frac{2\sqrt{13}+6}{2}$$

Divide 6+2\sqrt{13} by 2.

$$x=\sqrt{13}+3$$

Now solve the equation x=\frac{6±2\sqrt{13}}{2} when ± is minus. Subtract 2\sqrt{13} from 6.

$$x=\frac{6-2\sqrt{13}}{2}$$

Divide 6-2\sqrt{13} by 2.

$$x=3-\sqrt{13}$$

The equation is now solved.

$$x=\sqrt{13}+3 x=3-\sqrt{13}$$

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+1.

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

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

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

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

$$3x^{2}-3x-6=\left(x+1\right)\left(x+2\right)+x^{2}-4$$

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

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

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

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

Subtract 4 from 2 to get -2.

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

Subtract 2x^{2} from both sides.

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

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

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

Subtract 3x from both sides.

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

Combine -3x and -3x to get -6x.

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

Add 6 to both sides.

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

Add -2 and 6 to get 4.

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

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.

$$x^{2}-6x+\left(-3\right)^{2}=4+\left(-3\right)^{2}$$

Square -3.

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

Add 4 to 9.

$$x^{2}-6x+9=13$$

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}.

$$\left(x-3\right)^{2}=13$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{13}$$

Simplify.

$$x-3=\sqrt{13} x-3=-\sqrt{13}$$

Add 3 to both sides of the equation.

$$x=\sqrt{13}+3 x=3-\sqrt{13}$$