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

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

Solve for x

$$x=-\frac{2}{5}=-0.4$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -1,0,1 since division by zero is not defined. Multiply both sides of the equation by x\left(x-1\right)\left(x+1\right), the least common multiple of x^{2}-1,x+1,x^{2}-x.

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

Multiply x and x to get x^{2}.

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

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

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

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

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

Combine x^{2}\times 2 and 2x^{2} to get 4x^{2}.

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

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

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

Subtract x from both sides.

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

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

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

Add x^{2} to both sides.

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

Combine 4x^{2} and x^{2} to get 5x^{2}.

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

Subtract 2 from both sides.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 5x^{2}+ax+bx-2. To find a and b, set up a system to be solved.

$$a+b=-3 ab=5\left(-2\right)=-10$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -10.

$$1,-10 2,-5$$

Calculate the sum for each pair.

$$1-10=-9 2-5=-3$$

The solution is the pair that gives sum -3.

$$a=-5 b=2$$

Rewrite 5x^{2}-3x-2 as \left(5x^{2}-5x\right)+\left(2x-2\right).

$$\left(5x^{2}-5x\right)+\left(2x-2\right)$$

Factor out 5x in the first and 2 in the second group.

$$5x\left(x-1\right)+2\left(x-1\right)$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(5x+2\right)$$

To find equation solutions, solve x-1=0 and 5x+2=0.

$$x=1 x=-\frac{2}{5}$$

Variable x cannot be equal to 1.

$$x=-\frac{2}{5}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -1,0,1 since division by zero is not defined. Multiply both sides of the equation by x\left(x-1\right)\left(x+1\right), the least common multiple of x^{2}-1,x+1,x^{2}-x.

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

Multiply x and x to get x^{2}.

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

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

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

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

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

Combine x^{2}\times 2 and 2x^{2} to get 4x^{2}.

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

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

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

Subtract x from both sides.

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

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

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

Add x^{2} to both sides.

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

Combine 4x^{2} and x^{2} to get 5x^{2}.

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

Subtract 2 from both sides.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -3 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 5\left(-2\right)}}{2\times 5}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 5\left(-2\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-3\right)±\sqrt{9-20\left(-2\right)}}{2\times 5}$$

Multiply -20 times -2.

$$x=\frac{-\left(-3\right)±\sqrt{9+40}}{2\times 5}$$

Add 9 to 40.

$$x=\frac{-\left(-3\right)±\sqrt{49}}{2\times 5}$$

Take the square root of 49.

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

The opposite of -3 is 3.

$$x=\frac{3±7}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{3±7}{10}$$

Now solve the equation x=\frac{3±7}{10} when ± is plus. Add 3 to 7.

$$x=\frac{10}{10}$$

Divide 10 by 10.

$$x=1$$

Now solve the equation x=\frac{3±7}{10} when ± is minus. Subtract 7 from 3.

$$x=-\frac{4}{10}$$

Reduce the fraction \frac{-4}{10} to lowest terms by extracting and canceling out 2.

$$x=-\frac{2}{5}$$

The equation is now solved.

$$x=1 x=-\frac{2}{5}$$

Variable x cannot be equal to 1.

$$x=-\frac{2}{5}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -1,0,1 since division by zero is not defined. Multiply both sides of the equation by x\left(x-1\right)\left(x+1\right), the least common multiple of x^{2}-1,x+1,x^{2}-x.

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

Multiply x and x to get x^{2}.

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

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

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

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

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

Combine x^{2}\times 2 and 2x^{2} to get 4x^{2}.

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

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

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

Subtract x from both sides.

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

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

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

Add x^{2} to both sides.

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

Combine 4x^{2} and x^{2} to get 5x^{2}.

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

Divide both sides by 5.

$$\frac{5x^{2}-3x}{5}=\frac{2}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}-\frac{3}{5}x=\frac{2}{5}$$

Divide -\frac{3}{5}, the coefficient of the x term, by 2 to get -\frac{3}{10}. Then add the square of -\frac{3}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3}{5}x+\left(-\frac{3}{10}\right)^{2}=\frac{2}{5}+\left(-\frac{3}{10}\right)^{2}$$

Square -\frac{3}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{5}x+\frac{9}{100}=\frac{2}{5}+\frac{9}{100}$$

Add \frac{2}{5} to \frac{9}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3}{5}x+\frac{9}{100}=\frac{49}{100}$$

Factor x^{2}-\frac{3}{5}x+\frac{9}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{10}\right)^{2}=\frac{49}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{10}\right)^{2}}=\sqrt{\frac{49}{100}}$$

Simplify.

$$x-\frac{3}{10}=\frac{7}{10} x-\frac{3}{10}=-\frac{7}{10}$$

Add \frac{3}{10} to both sides of the equation.

$$x=1 x=-\frac{2}{5}$$

Variable x cannot be equal to 1.

$$x=-\frac{2}{5}$$