Solve 1+n-m/m-2n+m^2-n^2/m^2-2mn+m^2

$$1 + \frac { n - m } { m - 2 n } + \frac { m ^ { 2 } - n ^ { 2 } } { m ^ { 2 } - 2 m n + m ^ { 2 } }$$

Evaluate

$$\frac{m^{2}-3mn-2n^{2}}{2m\left(m-2n\right)}$$

Short Solution Steps

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

$$1+\frac{n-m}{m-2n}+\frac{m^{2}-n^{2}}{2m^{2}-2mn}$$

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

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

Cancel out m-n in both numerator and denominator.

$$1+\frac{n-m}{m-2n}+\frac{m+n}{2m}$$

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

$$\frac{m-2n}{m-2n}+\frac{n-m}{m-2n}+\frac{m+n}{2m}$$

Since \frac{m-2n}{m-2n} and \frac{n-m}{m-2n} have the same denominator, add them by adding their numerators.

$$\frac{m-2n+n-m}{m-2n}+\frac{m+n}{2m}$$

Combine like terms in m-2n+n-m.

$$\frac{-n}{m-2n}+\frac{m+n}{2m}$$

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

$$\frac{-n\times 2m}{2m\left(m-2n\right)}+\frac{\left(m+n\right)\left(m-2n\right)}{2m\left(m-2n\right)}$$

Since \frac{-n\times 2m}{2m\left(m-2n\right)} and \frac{\left(m+n\right)\left(m-2n\right)}{2m\left(m-2n\right)} have the same denominator, add them by adding their numerators.

$$\frac{-n\times 2m+\left(m+n\right)\left(m-2n\right)}{2m\left(m-2n\right)}$$

Do the multiplications in -n\times 2m+\left(m+n\right)\left(m-2n\right).

$$\frac{-2nm+m^{2}-2mn+nm-2n^{2}}{2m\left(m-2n\right)}$$

Combine like terms in -2nm+m^{2}-2mn+nm-2n^{2}.

$$\frac{m^{2}-3nm-2n^{2}}{2m\left(m-2n\right)}$$

Expand 2m\left(m-2n\right).

$$\frac{m^{2}-3nm-2n^{2}}{2m^{2}-4mn}$$

Expand

$$\frac{m^{2}-3mn-2n^{2}}{2m\left(m-2n\right)}$$

Short Solution Steps

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

$$1+\frac{n-m}{m-2n}+\frac{m^{2}-n^{2}}{2m^{2}-2mn}$$

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

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

Cancel out m-n in both numerator and denominator.

$$1+\frac{n-m}{m-2n}+\frac{m+n}{2m}$$

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

$$\frac{m-2n}{m-2n}+\frac{n-m}{m-2n}+\frac{m+n}{2m}$$

Since \frac{m-2n}{m-2n} and \frac{n-m}{m-2n} have the same denominator, add them by adding their numerators.

$$\frac{m-2n+n-m}{m-2n}+\frac{m+n}{2m}$$

Combine like terms in m-2n+n-m.

$$\frac{-n}{m-2n}+\frac{m+n}{2m}$$

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

$$\frac{-n\times 2m}{2m\left(m-2n\right)}+\frac{\left(m+n\right)\left(m-2n\right)}{2m\left(m-2n\right)}$$

Since \frac{-n\times 2m}{2m\left(m-2n\right)} and \frac{\left(m+n\right)\left(m-2n\right)}{2m\left(m-2n\right)} have the same denominator, add them by adding their numerators.

$$\frac{-n\times 2m+\left(m+n\right)\left(m-2n\right)}{2m\left(m-2n\right)}$$

Do the multiplications in -n\times 2m+\left(m+n\right)\left(m-2n\right).

$$\frac{-2nm+m^{2}-2mn+nm-2n^{2}}{2m\left(m-2n\right)}$$

Combine like terms in -2nm+m^{2}-2mn+nm-2n^{2}.

$$\frac{m^{2}-3nm-2n^{2}}{2m\left(m-2n\right)}$$

Expand 2m\left(m-2n\right).

$$\frac{m^{2}-3nm-2n^{2}}{2m^{2}-4mn}$$