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

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

Solve for x

$$x = \frac{3 \sqrt{2}}{2} \approx 2.121320344$$

$$x = -\frac{3 \sqrt{2}}{2} \approx -2.121320344$$

Steps by Finding Square Root

Variable x cannot be equal to any of the values -2,-1,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+1\right)\left(x+2\right), the least common multiple of x^{3}+2x^{2}-x-2,1-x,x+1.

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

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

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

Add 1 and 2 to get 3.

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

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

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

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

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

Subtract 3x^{2} from both sides.

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

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

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

Subtract 3x from both sides.

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

Combine 3x and -3x to get 0.

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

Subtract 3 from both sides.

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

Subtract 3 from -6 to get -9.

$$-2x^{2}=-9$$

Divide both sides by -2.

$$x^{2}=\frac{-9}{-2}$$

Fraction \frac{-9}{-2} can be simplified to \frac{9}{2} by removing the negative sign from both the numerator and the denominator.

$$x^{2}=\frac{9}{2}$$

Take the square root of both sides of the equation.

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

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,-1,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+1\right)\left(x+2\right), the least common multiple of x^{3}+2x^{2}-x-2,1-x,x+1.

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

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

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

Add 1 and 2 to get 3.

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

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

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

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

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

Subtract 3x^{2} from both sides.

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

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

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

Subtract 3x from both sides.

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

Combine 3x and -3x to get 0.

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

Add 6 to both sides.

$$3-2x^{2}+6=0$$

Add 3 and 6 to get 9.

$$9-2x^{2}=0$$

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.

$$-2x^{2}+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 0 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-2\right)\times 9}}{2\left(-2\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-2\right)\times 9}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{0±\sqrt{8\times 9}}{2\left(-2\right)}$$

Multiply 8 times 9.

$$x=\frac{0±\sqrt{72}}{2\left(-2\right)}$$

Take the square root of 72.

$$x=\frac{0±6\sqrt{2}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{0±6\sqrt{2}}{-4}$$

Now solve the equation x=\frac{0±6\sqrt{2}}{-4} when ± is plus.

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

Now solve the equation x=\frac{0±6\sqrt{2}}{-4} when ± is minus.

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

The equation is now solved.

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