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

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

Solve for x

$$x=\frac{1}{6}\approx 0.166666667$$

Steps for Solving Linear Equation

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2x-1 and 4 is 4\left(2x-1\right). Multiply \frac{2x^{2}}{2x-1} times \frac{4}{4}. Multiply \frac{1}{4} times \frac{2x-1}{2x-1}.

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

Since \frac{4\times 2x^{2}}{4\left(2x-1\right)} and \frac{2x-1}{4\left(2x-1\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 4\times 2x^{2}+2x-1.

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

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

$$x=\frac{8x^{2}+2x-1}{8x-4}$$

Subtract \frac{8x^{2}+2x-1}{8x-4} from both sides.

$$x-\frac{8x^{2}+2x-1}{8x-4}=0$$

Factor 8x-4.

$$x-\frac{8x^{2}+2x-1}{4\left(2x-1\right)}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{4\left(2x-1\right)}{4\left(2x-1\right)}.

$$\frac{x\times 4\left(2x-1\right)}{4\left(2x-1\right)}-\frac{8x^{2}+2x-1}{4\left(2x-1\right)}=0$$

Since \frac{x\times 4\left(2x-1\right)}{4\left(2x-1\right)} and \frac{8x^{2}+2x-1}{4\left(2x-1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\times 4\left(2x-1\right)-\left(8x^{2}+2x-1\right)}{4\left(2x-1\right)}=0$$

Do the multiplications in x\times 4\left(2x-1\right)-\left(8x^{2}+2x-1\right).

$$\frac{8x^{2}-4x-8x^{2}-2x+1}{4\left(2x-1\right)}=0$$

Combine like terms in 8x^{2}-4x-8x^{2}-2x+1.

$$\frac{-6x+1}{4\left(2x-1\right)}=0$$

Variable x cannot be equal to \frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by 4\left(2x-1\right).

$$-6x+1=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$-6x=-1$$

Divide both sides by -6.

$$x=\frac{-1}{-6}$$

Fraction \frac{-1}{-6} can be simplified to \frac{1}{6} by removing the negative sign from both the numerator and the denominator.

$$x=\frac{1}{6}$$