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

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

Solve for x

$$x=1$$

$$x=-1$$

Steps by Finding Square Root

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+2\right), the least common multiple of x,x+2.

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

Multiply x+2 and x+2 to get \left(x+2\right)^{2}.

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

Use the distributive property to multiply x+2 by 3.

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

Use the distributive property to multiply x by x-1.

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

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

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

Combine 3x and x to get 4x.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+2\right)^{2}.

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

Subtract x^{2} from both sides.

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

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

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

Subtract 4x from both sides.

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

Combine 4x and -4x to get 0.

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

Subtract 6 from both sides.

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

Subtract 6 from 4 to get -2.

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

Divide both sides by -2.

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

Divide -2 by -2 to get 1.

$$x^{2}=1$$

Take the square root of both sides of the equation.

$$x=1 x=-1$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+2\right), the least common multiple of x,x+2.

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

Multiply x+2 and x+2 to get \left(x+2\right)^{2}.

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

Use the distributive property to multiply x+2 by 3.

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

Use the distributive property to multiply x by x-1.

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

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

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

Combine 3x and x to get 4x.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+2\right)^{2}.

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

Subtract x^{2} from both sides.

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

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

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

Subtract 4x from both sides.

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

Combine 4x and -4x to get 0.

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

Subtract 4 from both sides.

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

Subtract 4 from 6 to get 2.

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

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

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

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

$$x=\frac{0±\sqrt{0^{2}-4\left(-2\right)\times 2}}{2\left(-2\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-2\right)\times 2}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{0±\sqrt{8\times 2}}{2\left(-2\right)}$$

Multiply 8 times 2.

$$x=\frac{0±\sqrt{16}}{2\left(-2\right)}$$

Take the square root of 16.

$$x=\frac{0±4}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{0±4}{-4}$$

Now solve the equation x=\frac{0±4}{-4} when ± is plus. Divide 4 by -4.

$$x=-1$$

Now solve the equation x=\frac{0±4}{-4} when ± is minus. Divide -4 by -4.

$$x=1$$

The equation is now solved.

$$x=-1 x=1$$