Solve 2x-1/x+1=2x/x-1+5/x

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

Solve for x

$$x=\frac{\sqrt{201}+1}{20}\approx 0.758872344$$

$$x=\frac{1-\sqrt{201}}{20}\approx -0.658872344$$

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.

$$x\left(x-1\right)\left(2x-1\right)=x\left(x+1\right)\times 2x+\left(x^{2}-1\right)\times 5$$

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

$$\left(x^{2}-x\right)\left(2x-1\right)=x\left(x+1\right)\times 2x+\left(x^{2}-1\right)\times 5$$

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

$$2x^{3}-3x^{2}+x=x\left(x+1\right)\times 2x+\left(x^{2}-1\right)\times 5$$

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

$$2x^{3}-3x^{2}+x=x^{2}\left(x+1\right)\times 2+\left(x^{2}-1\right)\times 5$$

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

$$2x^{3}-3x^{2}+x=\left(x^{3}+x^{2}\right)\times 2+\left(x^{2}-1\right)\times 5$$

Use the distributive property to multiply x^{3}+x^{2} by 2.

$$2x^{3}-3x^{2}+x=2x^{3}+2x^{2}+\left(x^{2}-1\right)\times 5$$

Use the distributive property to multiply x^{2}-1 by 5.

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

Combine 2x^{2} and 5x^{2} to get 7x^{2}.

$$2x^{3}-3x^{2}+x=2x^{3}+7x^{2}-5$$

Subtract 2x^{3} from both sides.

$$2x^{3}-3x^{2}+x-2x^{3}=7x^{2}-5$$

Combine 2x^{3} and -2x^{3} to get 0.

$$-3x^{2}+x=7x^{2}-5$$

Subtract 7x^{2} from both sides.

$$-3x^{2}+x-7x^{2}=-5$$

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

$$-10x^{2}+x=-5$$

Add 5 to both sides.

$$-10x^{2}+x+5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 1 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-10\right)\times 5}}{2\left(-10\right)}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-10\right)\times 5}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-1±\sqrt{1+40\times 5}}{2\left(-10\right)}$$

Multiply 40 times 5.

$$x=\frac{-1±\sqrt{1+200}}{2\left(-10\right)}$$

Add 1 to 200.

$$x=\frac{-1±\sqrt{201}}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-1±\sqrt{201}}{-20}$$

Now solve the equation x=\frac{-1±\sqrt{201}}{-20} when ± is plus. Add -1 to \sqrt{201}.

$$x=\frac{\sqrt{201}-1}{-20}$$

Divide -1+\sqrt{201} by -20.

$$x=\frac{1-\sqrt{201}}{20}$$

Now solve the equation x=\frac{-1±\sqrt{201}}{-20} when ± is minus. Subtract \sqrt{201} from -1.

$$x=\frac{-\sqrt{201}-1}{-20}$$

Divide -1-\sqrt{201} by -20.

$$x=\frac{\sqrt{201}+1}{20}$$

The equation is now solved.

$$x=\frac{1-\sqrt{201}}{20} x=\frac{\sqrt{201}+1}{20}$$

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.

$$x\left(x-1\right)\left(2x-1\right)=x\left(x+1\right)\times 2x+\left(x^{2}-1\right)\times 5$$

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

$$\left(x^{2}-x\right)\left(2x-1\right)=x\left(x+1\right)\times 2x+\left(x^{2}-1\right)\times 5$$

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

$$2x^{3}-3x^{2}+x=x\left(x+1\right)\times 2x+\left(x^{2}-1\right)\times 5$$

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

$$2x^{3}-3x^{2}+x=x^{2}\left(x+1\right)\times 2+\left(x^{2}-1\right)\times 5$$

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

$$2x^{3}-3x^{2}+x=\left(x^{3}+x^{2}\right)\times 2+\left(x^{2}-1\right)\times 5$$

Use the distributive property to multiply x^{3}+x^{2} by 2.

$$2x^{3}-3x^{2}+x=2x^{3}+2x^{2}+\left(x^{2}-1\right)\times 5$$

Use the distributive property to multiply x^{2}-1 by 5.

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

Combine 2x^{2} and 5x^{2} to get 7x^{2}.

$$2x^{3}-3x^{2}+x=2x^{3}+7x^{2}-5$$

Subtract 2x^{3} from both sides.

$$2x^{3}-3x^{2}+x-2x^{3}=7x^{2}-5$$

Combine 2x^{3} and -2x^{3} to get 0.

$$-3x^{2}+x=7x^{2}-5$$

Subtract 7x^{2} from both sides.

$$-3x^{2}+x-7x^{2}=-5$$

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

$$-10x^{2}+x=-5$$

Divide both sides by -10.

$$\frac{-10x^{2}+x}{-10}=-\frac{5}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{1}{-10}x=-\frac{5}{-10}$$

Divide 1 by -10.

$$x^{2}-\frac{1}{10}x=-\frac{5}{-10}$$

Reduce the fraction \frac{-5}{-10} to lowest terms by extracting and canceling out 5.

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

Divide -\frac{1}{10}, the coefficient of the x term, by 2 to get -\frac{1}{20}. Then add the square of -\frac{1}{20} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{10}x+\left(-\frac{1}{20}\right)^{2}=\frac{1}{2}+\left(-\frac{1}{20}\right)^{2}$$

Square -\frac{1}{20} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{10}x+\frac{1}{400}=\frac{1}{2}+\frac{1}{400}$$

Add \frac{1}{2} to \frac{1}{400} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{10}x+\frac{1}{400}=\frac{201}{400}$$

Factor x^{2}-\frac{1}{10}x+\frac{1}{400}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{20}\right)^{2}=\frac{201}{400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{20}\right)^{2}}=\sqrt{\frac{201}{400}}$$

Simplify.

$$x-\frac{1}{20}=\frac{\sqrt{201}}{20} x-\frac{1}{20}=-\frac{\sqrt{201}}{20}$$

Add \frac{1}{20} to both sides of the equation.

$$x=\frac{\sqrt{201}+1}{20} x=\frac{1-\sqrt{201}}{20}$$