Solve x/x-1-x/x+1-x/x^2-2x+1=2/x^2-1-frac{1}{x-1}-frac{1}{x+1}

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

Solve for x

$$x=\frac{1}{3}\approx 0.333333333$$

$$x=2$$

Steps Using Factoring By Grouping

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)^{2}, the least common multiple of x-1,x+1,x^{2}-2x+1,x^{2}-1.

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

Use the distributive property to multiply x^{2}-1 by x.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

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

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

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

To find the opposite of x^{3}-2x^{2}+x, find the opposite of each term.

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

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

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

Combine -x and -x to get -2x.

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

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

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

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

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

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

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

Combine -2x and -x to get -3x.

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

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

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

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

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

Add -2 and 1 to get -1.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

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

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

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

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

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

Combine 2x and 2x to get 4x.

$$-3x+x^{2}=4x-1-2x^{2}-1$$

Subtract 1 from -1 to get -2.

$$-3x+x^{2}=4x-2-2x^{2}$$

Subtract 4x from both sides.

$$-3x+x^{2}-4x=-2-2x^{2}$$

Combine -3x and -4x to get -7x.

$$-7x+x^{2}=-2-2x^{2}$$

Subtract -2 from both sides.

$$-7x+x^{2}-\left(-2\right)=-2x^{2}$$

The opposite of -2 is 2.

$$-7x+x^{2}+2=-2x^{2}$$

Add 2x^{2} to both sides.

$$-7x+x^{2}+2+2x^{2}=0$$

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

$$-7x+3x^{2}+2=0$$

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

$$3x^{2}-7x+2=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 3x^{2}+ax+bx+2. To find a and b, set up a system to be solved.

$$a+b=-7 ab=3\times 2=6$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 6.

$$-1,-6 -2,-3$$

Calculate the sum for each pair.

$$-1-6=-7 -2-3=-5$$

The solution is the pair that gives sum -7.

$$a=-6 b=-1$$

Rewrite 3x^{2}-7x+2 as \left(3x^{2}-6x\right)+\left(-x+2\right).

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

Factor out 3x in the first and -1 in the second group.

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

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

$$\left(x-2\right)\left(3x-1\right)$$

To find equation solutions, solve x-2=0 and 3x-1=0.

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

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)^{2}, the least common multiple of x-1,x+1,x^{2}-2x+1,x^{2}-1.

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

Use the distributive property to multiply x^{2}-1 by x.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

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

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

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

To find the opposite of x^{3}-2x^{2}+x, find the opposite of each term.

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

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

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

Combine -x and -x to get -2x.

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

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

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

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

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

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

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

Combine -2x and -x to get -3x.

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

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

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

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

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

Add -2 and 1 to get -1.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

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

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

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

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

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

Combine 2x and 2x to get 4x.

$$-3x+x^{2}=4x-1-2x^{2}-1$$

Subtract 1 from -1 to get -2.

$$-3x+x^{2}=4x-2-2x^{2}$$

Subtract 4x from both sides.

$$-3x+x^{2}-4x=-2-2x^{2}$$

Combine -3x and -4x to get -7x.

$$-7x+x^{2}=-2-2x^{2}$$

Subtract -2 from both sides.

$$-7x+x^{2}-\left(-2\right)=-2x^{2}$$

The opposite of -2 is 2.

$$-7x+x^{2}+2=-2x^{2}$$

Add 2x^{2} to both sides.

$$-7x+x^{2}+2+2x^{2}=0$$

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

$$-7x+3x^{2}+2=0$$

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.

$$3x^{2}-7x+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -7 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 3\times 2}}{2\times 3}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 3\times 2}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-7\right)±\sqrt{49-12\times 2}}{2\times 3}$$

Multiply -12 times 2.

$$x=\frac{-\left(-7\right)±\sqrt{49-24}}{2\times 3}$$

Add 49 to -24.

$$x=\frac{-\left(-7\right)±\sqrt{25}}{2\times 3}$$

Take the square root of 25.

$$x=\frac{-\left(-7\right)±5}{2\times 3}$$

The opposite of -7 is 7.

$$x=\frac{7±5}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{7±5}{6}$$

Now solve the equation x=\frac{7±5}{6} when ± is plus. Add 7 to 5.

$$x=\frac{12}{6}$$

Divide 12 by 6.

$$x=2$$

Now solve the equation x=\frac{7±5}{6} when ± is minus. Subtract 5 from 7.

$$x=\frac{2}{6}$$

Reduce the fraction \frac{2}{6} to lowest terms by extracting and canceling out 2.

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

The equation is now solved.

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

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)^{2}, the least common multiple of x-1,x+1,x^{2}-2x+1,x^{2}-1.

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

Use the distributive property to multiply x^{2}-1 by x.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

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

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

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

To find the opposite of x^{3}-2x^{2}+x, find the opposite of each term.

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

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

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

Combine -x and -x to get -2x.

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

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

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

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

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

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

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

Combine -2x and -x to get -3x.

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

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

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

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

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

Add -2 and 1 to get -1.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

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

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

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

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

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

Combine 2x and 2x to get 4x.

$$-3x+x^{2}=4x-1-2x^{2}-1$$

Subtract 1 from -1 to get -2.

$$-3x+x^{2}=4x-2-2x^{2}$$

Subtract 4x from both sides.

$$-3x+x^{2}-4x=-2-2x^{2}$$

Combine -3x and -4x to get -7x.

$$-7x+x^{2}=-2-2x^{2}$$

Add 2x^{2} to both sides.

$$-7x+x^{2}+2x^{2}=-2$$

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

$$-7x+3x^{2}=-2$$

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.

$$3x^{2}-7x=-2$$

Divide both sides by 3.

$$\frac{3x^{2}-7x}{3}=-\frac{2}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}-\frac{7}{3}x=-\frac{2}{3}$$

Divide -\frac{7}{3}, the coefficient of the x term, by 2 to get -\frac{7}{6}. Then add the square of -\frac{7}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{3}x+\left(-\frac{7}{6}\right)^{2}=-\frac{2}{3}+\left(-\frac{7}{6}\right)^{2}$$

Square -\frac{7}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=-\frac{2}{3}+\frac{49}{36}$$

Add -\frac{2}{3} to \frac{49}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=\frac{25}{36}$$

Factor x^{2}-\frac{7}{3}x+\frac{49}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{6}\right)^{2}=\frac{25}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{6}\right)^{2}}=\sqrt{\frac{25}{36}}$$

Simplify.

$$x-\frac{7}{6}=\frac{5}{6} x-\frac{7}{6}=-\frac{5}{6}$$

Add \frac{7}{6} to both sides of the equation.

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