Solve x/2-frac{1{x}}-1/8x+4+1/4

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{x}{x}.

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

Since \frac{2x}{x} and \frac{1}{x} have the same denominator, subtract them by subtracting their numerators.

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

Divide x by \frac{2x-1}{x} by multiplying x by the reciprocal of \frac{2x-1}{x}.

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

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

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

Factor 8x+4.

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

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

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

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

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

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

$$\frac{8x^{3}+4x^{2}-2x+1}{4\left(2x-1\right)\left(2x+1\right)}+\frac{1}{4}$$

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

$$\frac{8x^{3}+4x^{2}-2x+1}{4\left(2x-1\right)\left(2x+1\right)}+\frac{\left(2x-1\right)\left(2x+1\right)}{4\left(2x-1\right)\left(2x+1\right)}$$

Since \frac{8x^{3}+4x^{2}-2x+1}{4\left(2x-1\right)\left(2x+1\right)} and \frac{\left(2x-1\right)\left(2x+1\right)}{4\left(2x-1\right)\left(2x+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{8x^{3}+4x^{2}-2x+1+\left(2x-1\right)\left(2x+1\right)}{4\left(2x-1\right)\left(2x+1\right)}$$

Do the multiplications in 8x^{3}+4x^{2}-2x+1+\left(2x-1\right)\left(2x+1\right).

$$\frac{8x^{3}+4x^{2}-2x+1+4x^{2}+2x-2x-1}{4\left(2x-1\right)\left(2x+1\right)}$$

Combine like terms in 8x^{3}+4x^{2}-2x+1+4x^{2}+2x-2x-1.

$$\frac{8x^{3}+8x^{2}-2x}{4\left(2x-1\right)\left(2x+1\right)}$$

Factor the expressions that are not already factored in \frac{8x^{3}+8x^{2}-2x}{4\left(2x-1\right)\left(2x+1\right)}.

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

Cancel out 2\times 2 in both numerator and denominator.

$$\frac{2x\left(x-\left(-\frac{1}{2}\sqrt{2}-\frac{1}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{2}-\frac{1}{2}\right)\right)}{\left(2x-1\right)\left(2x+1\right)}$$

Expand \left(2x-1\right)\left(2x+1\right).

$$\frac{2x\left(x-\left(-\frac{1}{2}\sqrt{2}-\frac{1}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{2}-\frac{1}{2}\right)\right)}{4x^{2}-1}$$

To find the opposite of -\frac{1}{2}\sqrt{2}-\frac{1}{2}, find the opposite of each term.

$$\frac{2x\left(x-\left(-\frac{1}{2}\sqrt{2}\right)-\left(-\frac{1}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{2}-\frac{1}{2}\right)\right)}{4x^{2}-1}$$

The opposite of -\frac{1}{2}\sqrt{2} is \frac{1}{2}\sqrt{2}.

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

The opposite of -\frac{1}{2} is \frac{1}{2}.

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

To find the opposite of \frac{1}{2}\sqrt{2}-\frac{1}{2}, find the opposite of each term.

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

The opposite of -\frac{1}{2} is \frac{1}{2}.

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

Use the distributive property to multiply 2x by x+\frac{1}{2}\sqrt{2}+\frac{1}{2}.

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

Cancel out 2 and 2.

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

Cancel out 2 and 2.

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

Apply the distributive property by multiplying each term of 2x^{2}+x\sqrt{2}+x by each term of x-\frac{1}{2}\sqrt{2}+\frac{1}{2}.

$$\frac{2x^{3}+2x^{2}\left(-\frac{1}{2}\right)\sqrt{2}+2x^{2}\times \frac{1}{2}+\sqrt{2}x^{2}+x\sqrt{2}\left(-\frac{1}{2}\right)\sqrt{2}+x\sqrt{2}\times \frac{1}{2}+x^{2}+x\left(-\frac{1}{2}\right)\sqrt{2}+x\times \frac{1}{2}}{4x^{2}-1}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{2x^{3}+2x^{2}\left(-\frac{1}{2}\right)\sqrt{2}+2x^{2}\times \frac{1}{2}+\sqrt{2}x^{2}+x\times 2\left(-\frac{1}{2}\right)+x\sqrt{2}\times \frac{1}{2}+x^{2}+x\left(-\frac{1}{2}\right)\sqrt{2}+x\times \frac{1}{2}}{4x^{2}-1}$$

Cancel out 2 and 2.

$$\frac{2x^{3}-x^{2}\sqrt{2}+2x^{2}\times \frac{1}{2}+\sqrt{2}x^{2}+x\times 2\left(-\frac{1}{2}\right)+x\sqrt{2}\times \frac{1}{2}+x^{2}+x\left(-\frac{1}{2}\right)\sqrt{2}+x\times \frac{1}{2}}{4x^{2}-1}$$

Cancel out 2 and 2.

$$\frac{2x^{3}-x^{2}\sqrt{2}+x^{2}+\sqrt{2}x^{2}+x\times 2\left(-\frac{1}{2}\right)+x\sqrt{2}\times \frac{1}{2}+x^{2}+x\left(-\frac{1}{2}\right)\sqrt{2}+x\times \frac{1}{2}}{4x^{2}-1}$$

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

$$\frac{2x^{3}+x^{2}+x\times 2\left(-\frac{1}{2}\right)+x\sqrt{2}\times \frac{1}{2}+x^{2}+x\left(-\frac{1}{2}\right)\sqrt{2}+x\times \frac{1}{2}}{4x^{2}-1}$$

Cancel out 2 and 2.

$$\frac{2x^{3}+x^{2}+x\left(-1\right)+x\sqrt{2}\times \frac{1}{2}+x^{2}+x\left(-\frac{1}{2}\right)\sqrt{2}+x\times \frac{1}{2}}{4x^{2}-1}$$

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

$$\frac{2x^{3}+2x^{2}+x\left(-1\right)+x\sqrt{2}\times \frac{1}{2}+x\left(-\frac{1}{2}\right)\sqrt{2}+x\times \frac{1}{2}}{4x^{2}-1}$$

Combine x\sqrt{2}\times \frac{1}{2} and x\left(-\frac{1}{2}\right)\sqrt{2} to get 0.

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

Combine x\left(-1\right) and x\times \frac{1}{2} to get -\frac{1}{2}x.

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

Factor

$$\frac{2x\left(x-\frac{-\sqrt{2}-1}{2}\right)\left(x-\frac{\sqrt{2}-1}{2}\right)}{4x^{2}-1}$$