Solve 1/x-1+2/x+1=10x-2/x^2-x

$$\frac { 1 } { x - 1 } + \frac { 2 } { x + 1 } = \frac { 10 x - 2 } { x ^ { 2 } - x }$$

Solve for x

$$x=\frac{\sqrt{137}-9}{14}\approx 0.193192851$$

$$x=\frac{-\sqrt{137}-9}{14}\approx -1.478907136$$

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-1,x+1,x^{2}-x.

$$x\left(x+1\right)+x\left(x-1\right)\times 2=\left(x+1\right)\left(10x-2\right)$$

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

$$x^{2}+x+x\left(x-1\right)\times 2=\left(x+1\right)\left(10x-2\right)$$

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

$$x^{2}+x+\left(x^{2}-x\right)\times 2=\left(x+1\right)\left(10x-2\right)$$

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

$$x^{2}+x+2x^{2}-2x=\left(x+1\right)\left(10x-2\right)$$

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

$$3x^{2}+x-2x=\left(x+1\right)\left(10x-2\right)$$

Combine x and -2x to get -x.

$$3x^{2}-x=\left(x+1\right)\left(10x-2\right)$$

Use the distributive property to multiply x+1 by 10x-2 and combine like terms.

$$3x^{2}-x=10x^{2}+8x-2$$

Subtract 10x^{2} from both sides.

$$3x^{2}-x-10x^{2}=8x-2$$

Combine 3x^{2} and -10x^{2} to get -7x^{2}.

$$-7x^{2}-x=8x-2$$

Subtract 8x from both sides.

$$-7x^{2}-x-8x=-2$$

Combine -x and -8x to get -9x.

$$-7x^{2}-9x=-2$$

Add 2 to both sides.

$$-7x^{2}-9x+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, -9 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-7\right)\times 2}}{2\left(-7\right)}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\left(-7\right)\times 2}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-\left(-9\right)±\sqrt{81+28\times 2}}{2\left(-7\right)}$$

Multiply 28 times 2.

$$x=\frac{-\left(-9\right)±\sqrt{81+56}}{2\left(-7\right)}$$

Add 81 to 56.

$$x=\frac{-\left(-9\right)±\sqrt{137}}{2\left(-7\right)}$$

The opposite of -9 is 9.

$$x=\frac{9±\sqrt{137}}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{9±\sqrt{137}}{-14}$$

Now solve the equation x=\frac{9±\sqrt{137}}{-14} when ± is plus. Add 9 to \sqrt{137}.

$$x=\frac{\sqrt{137}+9}{-14}$$

Divide 9+\sqrt{137} by -14.

$$x=\frac{-\sqrt{137}-9}{14}$$

Now solve the equation x=\frac{9±\sqrt{137}}{-14} when ± is minus. Subtract \sqrt{137} from 9.

$$x=\frac{9-\sqrt{137}}{-14}$$

Divide 9-\sqrt{137} by -14.

$$x=\frac{\sqrt{137}-9}{14}$$

The equation is now solved.

$$x=\frac{-\sqrt{137}-9}{14} x=\frac{\sqrt{137}-9}{14}$$

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-1,x+1,x^{2}-x.

$$x\left(x+1\right)+x\left(x-1\right)\times 2=\left(x+1\right)\left(10x-2\right)$$

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

$$x^{2}+x+x\left(x-1\right)\times 2=\left(x+1\right)\left(10x-2\right)$$

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

$$x^{2}+x+\left(x^{2}-x\right)\times 2=\left(x+1\right)\left(10x-2\right)$$

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

$$x^{2}+x+2x^{2}-2x=\left(x+1\right)\left(10x-2\right)$$

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

$$3x^{2}+x-2x=\left(x+1\right)\left(10x-2\right)$$

Combine x and -2x to get -x.

$$3x^{2}-x=\left(x+1\right)\left(10x-2\right)$$

Use the distributive property to multiply x+1 by 10x-2 and combine like terms.

$$3x^{2}-x=10x^{2}+8x-2$$

Subtract 10x^{2} from both sides.

$$3x^{2}-x-10x^{2}=8x-2$$

Combine 3x^{2} and -10x^{2} to get -7x^{2}.

$$-7x^{2}-x=8x-2$$

Subtract 8x from both sides.

$$-7x^{2}-x-8x=-2$$

Combine -x and -8x to get -9x.

$$-7x^{2}-9x=-2$$

Divide both sides by -7.

$$\frac{-7x^{2}-9x}{-7}=-\frac{2}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$x^{2}+\left(-\frac{9}{-7}\right)x=-\frac{2}{-7}$$

Divide -9 by -7.

$$x^{2}+\frac{9}{7}x=-\frac{2}{-7}$$

Divide -2 by -7.

$$x^{2}+\frac{9}{7}x=\frac{2}{7}$$

Divide \frac{9}{7}, the coefficient of the x term, by 2 to get \frac{9}{14}. Then add the square of \frac{9}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{9}{7}x+\left(\frac{9}{14}\right)^{2}=\frac{2}{7}+\left(\frac{9}{14}\right)^{2}$$

Square \frac{9}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{9}{7}x+\frac{81}{196}=\frac{2}{7}+\frac{81}{196}$$

Add \frac{2}{7} to \frac{81}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{9}{7}x+\frac{81}{196}=\frac{137}{196}$$

Factor x^{2}+\frac{9}{7}x+\frac{81}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{9}{14}\right)^{2}=\frac{137}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{9}{14}\right)^{2}}=\sqrt{\frac{137}{196}}$$

Simplify.

$$x+\frac{9}{14}=\frac{\sqrt{137}}{14} x+\frac{9}{14}=-\frac{\sqrt{137}}{14}$$

Subtract \frac{9}{14} from both sides of the equation.

$$x=\frac{\sqrt{137}-9}{14} x=\frac{-\sqrt{137}-9}{14}$$