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

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

Solve for x

$$x = \frac{\sqrt{53} + 3}{2} \approx 5.140054945$$

$$x=\frac{3-\sqrt{53}}{2}\approx -2.140054945$$

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.

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

Multiply -1 and 2 to get -2.

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

Use the distributive property to multiply -2 by 1+x.

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

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

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

To find the opposite of -4-6x-2x^{2}, find the opposite of each term.

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

Add 1 and 4 to get 5.

$$5+6x+2x^{2}=\left(x-1\right)\left(x+2\right)\times 3$$

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

$$5+6x+2x^{2}=\left(x^{2}+x-2\right)\times 3$$

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

$$5+6x+2x^{2}=3x^{2}+3x-6$$

Subtract 3x^{2} from both sides.

$$5+6x+2x^{2}-3x^{2}=3x-6$$

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

$$5+6x-x^{2}=3x-6$$

Subtract 3x from both sides.

$$5+6x-x^{2}-3x=-6$$

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

$$5+3x-x^{2}=-6$$

Add 6 to both sides.

$$5+3x-x^{2}+6=0$$

Add 5 and 6 to get 11.

$$11+3x-x^{2}=0$$

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.

$$-x^{2}+3x+11=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 3 for b, and 11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-1\right)\times 11}}{2\left(-1\right)}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-1\right)\times 11}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-3±\sqrt{9+4\times 11}}{2\left(-1\right)}$$

Multiply 4 times 11.

$$x=\frac{-3±\sqrt{9+44}}{2\left(-1\right)}$$

Add 9 to 44.

$$x=\frac{-3±\sqrt{53}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-3±\sqrt{53}}{-2}$$

Now solve the equation x=\frac{-3±\sqrt{53}}{-2} when ± is plus. Add -3 to \sqrt{53}.

$$x=\frac{\sqrt{53}-3}{-2}$$

Divide -3+\sqrt{53} by -2.

$$x=\frac{3-\sqrt{53}}{2}$$

Now solve the equation x=\frac{-3±\sqrt{53}}{-2} when ± is minus. Subtract \sqrt{53} from -3.

$$x=\frac{-\sqrt{53}-3}{-2}$$

Divide -3-\sqrt{53} by -2.

$$x=\frac{\sqrt{53}+3}{2}$$

The equation is now solved.

$$x=\frac{3-\sqrt{53}}{2} x=\frac{\sqrt{53}+3}{2}$$

Steps for Completing the Square

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.

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

Multiply -1 and 2 to get -2.

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

Use the distributive property to multiply -2 by 1+x.

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

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

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

To find the opposite of -4-6x-2x^{2}, find the opposite of each term.

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

Add 1 and 4 to get 5.

$$5+6x+2x^{2}=\left(x-1\right)\left(x+2\right)\times 3$$

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

$$5+6x+2x^{2}=\left(x^{2}+x-2\right)\times 3$$

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

$$5+6x+2x^{2}=3x^{2}+3x-6$$

Subtract 3x^{2} from both sides.

$$5+6x+2x^{2}-3x^{2}=3x-6$$

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

$$5+6x-x^{2}=3x-6$$

Subtract 3x from both sides.

$$5+6x-x^{2}-3x=-6$$

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

$$5+3x-x^{2}=-6$$

Subtract 5 from both sides.

$$3x-x^{2}=-6-5$$

Subtract 5 from -6 to get -11.

$$3x-x^{2}=-11$$

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.

$$-x^{2}+3x=-11$$

Divide both sides by -1.

$$\frac{-x^{2}+3x}{-1}=-\frac{11}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{3}{-1}x=-\frac{11}{-1}$$

Divide 3 by -1.

$$x^{2}-3x=-\frac{11}{-1}$$

Divide -11 by -1.

$$x^{2}-3x=11$$

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

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=11+\left(-\frac{3}{2}\right)^{2}$$

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

$$x^{2}-3x+\frac{9}{4}=11+\frac{9}{4}$$

Add 11 to \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{53}{4}$$

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

$$\left(x-\frac{3}{2}\right)^{2}=\frac{53}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{53}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{\sqrt{53}}{2} x-\frac{3}{2}=-\frac{\sqrt{53}}{2}$$

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

$$x=\frac{\sqrt{53}+3}{2} x=\frac{3-\sqrt{53}}{2}$$