Solve (m+n)/(m-n)-((m)^(2)+(n)^(2))/((m)^(2)-(n)^(2)) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

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

Evaluate

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

Short Solution Steps

Factor $m^{2}-n^{2}$.

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

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

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

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

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

Do the multiplications in $\left(m+n\right)\left(m+n\right)-\left(m^{2}+n^{2}\right)$.

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

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

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

Expand $\left(m+n\right)\left(m-n\right)$.

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

Expand

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

Short Solution Steps

Factor $m^{2}-n^{2}$.

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

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

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

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

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

Do the multiplications in $\left(m+n\right)\left(m+n\right)-\left(m^{2}+n^{2}\right)$.

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

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

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

Expand $\left(m+n\right)\left(m-n\right)$.

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