Solve x-x/x^3+frac{x^2{x-frac{x^2/x-frac{1{x}}}}}{1-x/x+frac{1{x}}-2}

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

Evaluate

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

Short Solution Steps

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

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

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

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

Do the multiplications in xx-1.

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

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

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

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

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

Factor x^{2}-1.

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

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

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

Since \frac{x\left(x-1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)} and \frac{x^{3}}{\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

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

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

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

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

Divide x^{2} by \frac{-x}{\left(x-1\right)\left(x+1\right)} by multiplying x^{2} by the reciprocal of \frac{-x}{\left(x-1\right)\left(x+1\right)}.

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

Cancel out x in both numerator and denominator.

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

Anything divided by -1 gives its opposite.

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

Factor x^{3}-x\left(x-1\right)\left(x+1\right).

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

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

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

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

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

Do the multiplications in xx-x.

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

Factor the expressions that are not already factored in \frac{x^{2}-x}{x}.

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

Cancel out x in both numerator and denominator.

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

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

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

Since \frac{xx}{x} and \frac{1}{x} have the same denominator, add them by adding their numerators.

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

Do the multiplications in xx+1.

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

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

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

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

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

Subtract 2 from 1 to get -1.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply -1 times \frac{x^{2}+1}{x^{2}+1}.

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

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

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

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

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

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

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

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

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

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

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

Expand

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

Short Solution Steps

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

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

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

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

Do the multiplications in xx-1.

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

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

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

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

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

Factor x^{2}-1.

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

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

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

Since \frac{x\left(x-1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)} and \frac{x^{3}}{\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

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

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

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

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

Divide x^{2} by \frac{-x}{\left(x-1\right)\left(x+1\right)} by multiplying x^{2} by the reciprocal of \frac{-x}{\left(x-1\right)\left(x+1\right)}.

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

Cancel out x in both numerator and denominator.

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

Anything divided by -1 gives its opposite.

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

Factor x^{3}-x\left(x-1\right)\left(x+1\right).

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

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

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

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

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

Do the multiplications in xx-x.

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

Factor the expressions that are not already factored in \frac{x^{2}-x}{x}.

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

Cancel out x in both numerator and denominator.

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

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

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

Since \frac{xx}{x} and \frac{1}{x} have the same denominator, add them by adding their numerators.

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

Do the multiplications in xx+1.

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

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

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

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

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

Subtract 2 from 1 to get -1.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply -1 times \frac{x^{2}+1}{x^{2}+1}.

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

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

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

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

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

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

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

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

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

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

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