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

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

Solve for x

$$x=2$$

Steps Using Factoring By Grouping

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

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

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

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

To find the opposite of x^{2}-x+1, find the opposite of each term.

$$2x+2-x^{2}+x-1=2x-1$$

Combine 2x and x to get 3x.

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

Subtract 1 from 2 to get 1.

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

Subtract 2x from both sides.

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

Combine 3x and -2x to get x.

$$x+1-x^{2}=-1$$

Add 1 to both sides.

$$x+1-x^{2}+1=0$$

Add 1 and 1 to get 2.

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

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+2. To find a and b, set up a system to be solved.

$$a+b=1 ab=-2=-2$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. The only such pair is the system solution.

$$a=2 b=-1$$

Rewrite -x^{2}+x+2 as \left(-x^{2}+2x\right)+\left(-x+2\right).

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

Factor out -x in the first and -1 in the second group.

$$-x\left(x-2\right)-\left(x-2\right)$$

Factor out common term x-2 by using distributive property.

$$\left(x-2\right)\left(-x-1\right)$$

To find equation solutions, solve x-2=0 and -x-1=0.

$$x=2 x=-1$$

Variable x cannot be equal to -1.

$$x=2$$

Steps Using the Quadratic Formula

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

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

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

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

To find the opposite of x^{2}-x+1, find the opposite of each term.

$$2x+2-x^{2}+x-1=2x-1$$

Combine 2x and x to get 3x.

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

Subtract 1 from 2 to get 1.

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

Subtract 2x from both sides.

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

Combine 3x and -2x to get x.

$$x+1-x^{2}=-1$$

Add 1 to both sides.

$$x+1-x^{2}+1=0$$

Add 1 and 1 to get 2.

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

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.

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

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

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

Square 1.

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

Multiply -4 times -1.

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

Multiply 4 times 2.

$$x=\frac{-1±\sqrt{1+8}}{2\left(-1\right)}$$

Add 1 to 8.

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

Take the square root of 9.

$$x=\frac{-1±3}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-1±3}{-2}$$

Now solve the equation x=\frac{-1±3}{-2} when ± is plus. Add -1 to 3.

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

Divide 2 by -2.

$$x=-1$$

Now solve the equation x=\frac{-1±3}{-2} when ± is minus. Subtract 3 from -1.

$$x=-\frac{4}{-2}$$

Divide -4 by -2.

$$x=2$$

The equation is now solved.

$$x=-1 x=2$$

Variable x cannot be equal to -1.

$$x=2$$

Steps for Completing the Square

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

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

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

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

To find the opposite of x^{2}-x+1, find the opposite of each term.

$$2x+2-x^{2}+x-1=2x-1$$

Combine 2x and x to get 3x.

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

Subtract 1 from 2 to get 1.

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

Subtract 2x from both sides.

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

Combine 3x and -2x to get x.

$$x+1-x^{2}=-1$$

Subtract 1 from both sides.

$$x-x^{2}=-1-1$$

Subtract 1 from -1 to get -2.

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

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.

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide 1 by -1.

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

Divide -2 by -1.

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

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

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

Add 2 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{9}{4}$$

Factor x^{2}-x+\frac{1}{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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

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

Add \frac{1}{2} to both sides of the equation.

$$x=2 x=-1$$

Variable x cannot be equal to -1.

$$x=2$$