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

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-3 and 2x-1 is \left(x-3\right)\left(2x-1\right). Multiply \frac{1}{x-3} times \frac{2x-1}{2x-1}. Multiply \frac{x^{2}}{2x-1} times \frac{x-3}{x-3}.

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

Since \frac{2x-1}{\left(x-3\right)\left(2x-1\right)} and \frac{x^{2}\left(x-3\right)}{\left(x-3\right)\left(2x-1\right)} have the same denominator, subtract them by subtracting their numerators.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and x-2 is x\left(x-2\right). Multiply \frac{x-2}{x} times \frac{x-2}{x-2}. Multiply \frac{x-3}{x-2} times \frac{x}{x}.

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

Since \frac{\left(x-2\right)\left(x-2\right)}{x\left(x-2\right)} and \frac{\left(x-3\right)x}{x\left(x-2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x-2\right)\left(x-2\right)-\left(x-3\right)x.

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

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

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

Divide \frac{2x-1-x^{3}+3x^{2}}{\left(x-3\right)\left(2x-1\right)} by \frac{-x+4}{x\left(x-2\right)} by multiplying \frac{2x-1-x^{3}+3x^{2}}{\left(x-3\right)\left(2x-1\right)} by the reciprocal of \frac{-x+4}{x\left(x-2\right)}.

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

Use the distributive property to multiply 2x-1-x^{3}+3x^{2} by x.

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

Use the distributive property to multiply 2x^{2}-x-x^{4}+3x^{3} by x-2 and combine like terms.

$$\frac{-4x^{3}-5x^{2}+2x-x^{5}+5x^{4}}{\left(x-3\right)\left(2x-1\right)\left(-x+4\right)}$$

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

$$\frac{-4x^{3}-5x^{2}+2x-x^{5}+5x^{4}}{\left(2x^{2}-7x+3\right)\left(-x+4\right)}$$

Use the distributive property to multiply 2x^{2}-7x+3 by -x+4 and combine like terms.

$$\frac{-4x^{3}-5x^{2}+2x-x^{5}+5x^{4}}{-2x^{3}+15x^{2}-31x+12}$$

Expand

$$\frac{2x-5x^{2}-4x^{3}+5x^{4}-x^{5}}{\left(4-x\right)\left(x-3\right)\left(2x-1\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-3 and 2x-1 is \left(x-3\right)\left(2x-1\right). Multiply \frac{1}{x-3} times \frac{2x-1}{2x-1}. Multiply \frac{x^{2}}{2x-1} times \frac{x-3}{x-3}.

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

Since \frac{2x-1}{\left(x-3\right)\left(2x-1\right)} and \frac{x^{2}\left(x-3\right)}{\left(x-3\right)\left(2x-1\right)} have the same denominator, subtract them by subtracting their numerators.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and x-2 is x\left(x-2\right). Multiply \frac{x-2}{x} times \frac{x-2}{x-2}. Multiply \frac{x-3}{x-2} times \frac{x}{x}.

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

Since \frac{\left(x-2\right)\left(x-2\right)}{x\left(x-2\right)} and \frac{\left(x-3\right)x}{x\left(x-2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x-2\right)\left(x-2\right)-\left(x-3\right)x.

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

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

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

Divide \frac{2x-1-x^{3}+3x^{2}}{\left(x-3\right)\left(2x-1\right)} by \frac{-x+4}{x\left(x-2\right)} by multiplying \frac{2x-1-x^{3}+3x^{2}}{\left(x-3\right)\left(2x-1\right)} by the reciprocal of \frac{-x+4}{x\left(x-2\right)}.

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

Use the distributive property to multiply 2x-1-x^{3}+3x^{2} by x.

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

Use the distributive property to multiply 2x^{2}-x-x^{4}+3x^{3} by x-2 and combine like terms.

$$\frac{-4x^{3}-5x^{2}+2x-x^{5}+5x^{4}}{\left(x-3\right)\left(2x-1\right)\left(-x+4\right)}$$

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

$$\frac{-4x^{3}-5x^{2}+2x-x^{5}+5x^{4}}{\left(2x^{2}-7x+3\right)\left(-x+4\right)}$$

Use the distributive property to multiply 2x^{2}-7x+3 by -x+4 and combine like terms.

$$\frac{-4x^{3}-5x^{2}+2x-x^{5}+5x^{4}}{-2x^{3}+15x^{2}-31x+12}$$