Solve 4x/x+1+x/2x-1=2

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

Solve for x (complex solution)

$$x=\frac{\sqrt{15}i}{10}+\frac{1}{2}\approx 0.5+0.387298335i$$

$$x=-\frac{\sqrt{15}i}{10}+\frac{1}{2}\approx 0.5-0.387298335i$$

Steps Using the Quadratic Formula

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

$$\left(2x-1\right)\times 4x+\left(x+1\right)x=2\left(2x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 2x-1 by 4.

$$\left(8x-4\right)x+\left(x+1\right)x=2\left(2x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 8x-4 by x.

$$8x^{2}-4x+\left(x+1\right)x=2\left(2x-1\right)\left(x+1\right)$$

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

$$8x^{2}-4x+x^{2}+x=2\left(2x-1\right)\left(x+1\right)$$

Combine 8x^{2} and x^{2} to get 9x^{2}.

$$9x^{2}-4x+x=2\left(2x-1\right)\left(x+1\right)$$

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

$$9x^{2}-3x=2\left(2x-1\right)\left(x+1\right)$$

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

$$9x^{2}-3x=\left(4x-2\right)\left(x+1\right)$$

Use the distributive property to multiply 4x-2 by x+1 and combine like terms.

$$9x^{2}-3x=4x^{2}+2x-2$$

Subtract 4x^{2} from both sides.

$$9x^{2}-3x-4x^{2}=2x-2$$

Combine 9x^{2} and -4x^{2} to get 5x^{2}.

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

Subtract 2x from both sides.

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

Combine -3x and -2x to get -5x.

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

Add 2 to both sides.

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

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

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

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 5\times 2}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-5\right)±\sqrt{25-20\times 2}}{2\times 5}$$

Multiply -20 times 2.

$$x=\frac{-\left(-5\right)±\sqrt{25-40}}{2\times 5}$$

Add 25 to -40.

$$x=\frac{-\left(-5\right)±\sqrt{-15}}{2\times 5}$$

Take the square root of -15.

$$x=\frac{-\left(-5\right)±\sqrt{15}i}{2\times 5}$$

The opposite of -5 is 5.

$$x=\frac{5±\sqrt{15}i}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{5±\sqrt{15}i}{10}$$

Now solve the equation x=\frac{5±\sqrt{15}i}{10} when ± is plus. Add 5 to i\sqrt{15}.

$$x=\frac{5+\sqrt{15}i}{10}$$

Divide 5+i\sqrt{15} by 10.

$$x=\frac{\sqrt{15}i}{10}+\frac{1}{2}$$

Now solve the equation x=\frac{5±\sqrt{15}i}{10} when ± is minus. Subtract i\sqrt{15} from 5.

$$x=\frac{-\sqrt{15}i+5}{10}$$

Divide 5-i\sqrt{15} by 10.

$$x=-\frac{\sqrt{15}i}{10}+\frac{1}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{15}i}{10}+\frac{1}{2} x=-\frac{\sqrt{15}i}{10}+\frac{1}{2}$$

Steps for Completing the Square

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

$$\left(2x-1\right)\times 4x+\left(x+1\right)x=2\left(2x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 2x-1 by 4.

$$\left(8x-4\right)x+\left(x+1\right)x=2\left(2x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 8x-4 by x.

$$8x^{2}-4x+\left(x+1\right)x=2\left(2x-1\right)\left(x+1\right)$$

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

$$8x^{2}-4x+x^{2}+x=2\left(2x-1\right)\left(x+1\right)$$

Combine 8x^{2} and x^{2} to get 9x^{2}.

$$9x^{2}-4x+x=2\left(2x-1\right)\left(x+1\right)$$

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

$$9x^{2}-3x=2\left(2x-1\right)\left(x+1\right)$$

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

$$9x^{2}-3x=\left(4x-2\right)\left(x+1\right)$$

Use the distributive property to multiply 4x-2 by x+1 and combine like terms.

$$9x^{2}-3x=4x^{2}+2x-2$$

Subtract 4x^{2} from both sides.

$$9x^{2}-3x-4x^{2}=2x-2$$

Combine 9x^{2} and -4x^{2} to get 5x^{2}.

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

Subtract 2x from both sides.

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

Combine -3x and -2x to get -5x.

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

Divide both sides by 5.

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

Dividing by 5 undoes the multiplication by 5.

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

Divide -5 by 5.

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

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.

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

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

$$x^{2}-x+\frac{1}{4}=-\frac{2}{5}+\frac{1}{4}$$

Add -\frac{2}{5} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-x+\frac{1}{4}=-\frac{3}{20}$$

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}.

$$\left(x-\frac{1}{2}\right)^{2}=-\frac{3}{20}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{-\frac{3}{20}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{15}i}{10} x-\frac{1}{2}=-\frac{\sqrt{15}i}{10}$$

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

$$x=\frac{\sqrt{15}i}{10}+\frac{1}{2} x=-\frac{\sqrt{15}i}{10}+\frac{1}{2}$$