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

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

Evaluate

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

Short Solution Steps

Factor x^{3}+x^{2}.

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

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

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

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

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

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

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

Combine like terms in 2x+2-1.

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

Divide \frac{3-2x}{x^{3}} by \frac{2x+1}{\left(x+1\right)x^{2}} by multiplying \frac{3-2x}{x^{3}} by the reciprocal of \frac{2x+1}{\left(x+1\right)x^{2}}.

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

Cancel out x^{2} in both numerator and denominator.

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

Use the distributive property to multiply x+1 by -2x+3 and combine like terms.

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

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

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

Expand

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

Short Solution Steps

Factor x^{3}+x^{2}.

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

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

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

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

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

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

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

Combine like terms in 2x+2-1.

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

Divide \frac{3-2x}{x^{3}} by \frac{2x+1}{\left(x+1\right)x^{2}} by multiplying \frac{3-2x}{x^{3}} by the reciprocal of \frac{2x+1}{\left(x+1\right)x^{2}}.

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

Cancel out x^{2} in both numerator and denominator.

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

Use the distributive property to multiply x+1 by -2x+3 and combine like terms.

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

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

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