Solve (1/4x^2-1/xy+1/y^2)(1/2x-y-frac{1}{y-2x}

$$( \frac{ 1 }{ 4 { x }^{ 2 } } - \frac{ 1 }{ xy } + \frac{ 1 }{ { y }^{ 2 } } )( \frac{ 1 }{ 2x-y } - \frac{ 1 }{ y-2x }$$

Evaluate

$$-\frac{y-2x}{2\left(xy\right)^{2}}$$

Short Solution Steps

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

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

Since \frac{y}{4yx^{2}} and \frac{4x}{4yx^{2}} have the same denominator, subtract them by subtracting their numerators.

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

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

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

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

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

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

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

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

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

Since \frac{-1}{-2x+y} and \frac{1}{-2x+y} have the same denominator, subtract them by subtracting their numerators. Subtract 1 from -1 to get -2.

$$\frac{y^{2}-4xy+4x^{2}}{4x^{2}y^{2}}\times \frac{-2}{-2x+y}$$

Multiply \frac{y^{2}-4xy+4x^{2}}{4x^{2}y^{2}} times \frac{-2}{-2x+y} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out 2 in both numerator and denominator.

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

Cancel out -1 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out -2x+y in both numerator and denominator.

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

Expand

$$-\frac{y-2x}{2\left(xy\right)^{2}}$$

Short Solution Steps

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

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

Since \frac{y}{4yx^{2}} and \frac{4x}{4yx^{2}} have the same denominator, subtract them by subtracting their numerators.

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

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

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

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

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

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

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

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

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

Since \frac{-1}{-2x+y} and \frac{1}{-2x+y} have the same denominator, subtract them by subtracting their numerators. Subtract 1 from -1 to get -2.

$$\frac{y^{2}-4xy+4x^{2}}{4x^{2}y^{2}}\times \frac{-2}{-2x+y}$$

Multiply \frac{y^{2}-4xy+4x^{2}}{4x^{2}y^{2}} times \frac{-2}{-2x+y} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out 2 in both numerator and denominator.

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

Cancel out -1 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out -2x+y in both numerator and denominator.

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