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

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

Solve for x

$$x=0$$

Steps Using Factoring

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

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

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

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

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

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

To find the opposite of x^{3}-1, find the opposite of each term.

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

Combine x^{3} and -x^{3} to get 0.

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

Add 1 and 1 to get 2.

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

Use the distributive property to multiply x-1 by 3.

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

Use the distributive property to multiply 3x-3 by x.

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

Swap sides so that all variable terms are on the left hand side.

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

Subtract 2 from both sides.

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

Subtract 2 from 2 to get 0.

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

Factor out x.

$$x\left(3x-3\right)=0$$

To find equation solutions, solve x=0 and 3x-3=0.

$$x=0 x=1$$

Variable x cannot be equal to 1.

$$x=0$$

Steps Using the Quadratic Formula

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

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

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

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

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

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

To find the opposite of x^{3}-1, find the opposite of each term.

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

Combine x^{3} and -x^{3} to get 0.

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

Add 1 and 1 to get 2.

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

Use the distributive property to multiply x-1 by 3.

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

Use the distributive property to multiply 3x-3 by x.

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

Swap sides so that all variable terms are on the left hand side.

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

Subtract 2 from both sides.

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

Subtract 2 from 2 to get 0.

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

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

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

Take the square root of \left(-3\right)^{2}.

$$x=\frac{-\left(-3\right)±3}{2\times 3}$$

The opposite of -3 is 3.

$$x=\frac{3±3}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{3±3}{6}$$

Now solve the equation x=\frac{3±3}{6} when ± is plus. Add 3 to 3.

$$x=\frac{6}{6}$$

Divide 6 by 6.

$$x=1$$

Now solve the equation x=\frac{3±3}{6} when ± is minus. Subtract 3 from 3.

$$x=\frac{0}{6}$$

Divide 0 by 6.

$$x=0$$

The equation is now solved.

$$x=1 x=0$$

Variable x cannot be equal to 1.

$$x=0$$

Steps for Completing the Square

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

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

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

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

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

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

To find the opposite of x^{3}-1, find the opposite of each term.

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

Combine x^{3} and -x^{3} to get 0.

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

Add 1 and 1 to get 2.

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

Use the distributive property to multiply x-1 by 3.

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

Use the distributive property to multiply 3x-3 by x.

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

Swap sides so that all variable terms are on the left hand side.

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

Subtract 2 from both sides.

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

Subtract 2 from 2 to get 0.

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

Divide both sides by 3.

$$\frac{3x^{2}-3x}{3}=\frac{0}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\left(-\frac{3}{3}\right)x=\frac{0}{3}$$

Divide -3 by 3.

$$x^{2}-x=\frac{0}{3}$$

Divide 0 by 3.

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

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.

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

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

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

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

$$x=1 x=0$$

Variable x cannot be equal to 1.

$$x=0$$