Solve r+3s/s+r-3s^2/s^2-r^2+r/s-r

$$\frac { r + 3 s } { s + r } - \frac { 3 s ^ { 2 } } { s ^ { 2 } - r ^ { 2 } } + \frac { r } { s - r }$$

Evaluate

$$-\frac{rs}{s^{2}-r^{2}}$$

Short Solution Steps

Factor s^{2}-r^{2}.

$$\frac{r+3s}{s+r}-\frac{3s^{2}}{\left(r+s\right)\left(-r+s\right)}+\frac{r}{s-r}$$

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

$$\frac{\left(r+3s\right)\left(-r+s\right)}{\left(r+s\right)\left(-r+s\right)}-\frac{3s^{2}}{\left(r+s\right)\left(-r+s\right)}+\frac{r}{s-r}$$

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

$$\frac{\left(r+3s\right)\left(-r+s\right)-3s^{2}}{\left(r+s\right)\left(-r+s\right)}+\frac{r}{s-r}$$

Do the multiplications in \left(r+3s\right)\left(-r+s\right)-3s^{2}.

$$\frac{-r^{2}+rs-3sr+3s^{2}-3s^{2}}{\left(r+s\right)\left(-r+s\right)}+\frac{r}{s-r}$$

Combine like terms in -r^{2}+rs-3sr+3s^{2}-3s^{2}.

$$\frac{-r^{2}-2rs}{\left(r+s\right)\left(-r+s\right)}+\frac{r}{s-r}$$

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

$$\frac{-r^{2}-2rs}{\left(r+s\right)\left(-r+s\right)}+\frac{r\left(r+s\right)}{\left(r+s\right)\left(-r+s\right)}$$

Since \frac{-r^{2}-2rs}{\left(r+s\right)\left(-r+s\right)} and \frac{r\left(r+s\right)}{\left(r+s\right)\left(-r+s\right)} have the same denominator, add them by adding their numerators.

$$\frac{-r^{2}-2rs+r\left(r+s\right)}{\left(r+s\right)\left(-r+s\right)}$$

Do the multiplications in -r^{2}-2rs+r\left(r+s\right).

$$\frac{-r^{2}-2rs+r^{2}+rs}{\left(r+s\right)\left(-r+s\right)}$$

Combine like terms in -r^{2}-2rs+r^{2}+rs.

$$\frac{-rs}{\left(r+s\right)\left(-r+s\right)}$$

Expand \left(r+s\right)\left(-r+s\right).

$$\frac{-rs}{-r^{2}+s^{2}}$$

Expand

$$-\frac{rs}{s^{2}-r^{2}}$$

Short Solution Steps

Factor s^{2}-r^{2}.

$$\frac{r+3s}{s+r}-\frac{3s^{2}}{\left(r+s\right)\left(-r+s\right)}+\frac{r}{s-r}$$

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

$$\frac{\left(r+3s\right)\left(-r+s\right)}{\left(r+s\right)\left(-r+s\right)}-\frac{3s^{2}}{\left(r+s\right)\left(-r+s\right)}+\frac{r}{s-r}$$

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

$$\frac{\left(r+3s\right)\left(-r+s\right)-3s^{2}}{\left(r+s\right)\left(-r+s\right)}+\frac{r}{s-r}$$

Do the multiplications in \left(r+3s\right)\left(-r+s\right)-3s^{2}.

$$\frac{-r^{2}+rs-3sr+3s^{2}-3s^{2}}{\left(r+s\right)\left(-r+s\right)}+\frac{r}{s-r}$$

Combine like terms in -r^{2}+rs-3sr+3s^{2}-3s^{2}.

$$\frac{-r^{2}-2rs}{\left(r+s\right)\left(-r+s\right)}+\frac{r}{s-r}$$

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

$$\frac{-r^{2}-2rs}{\left(r+s\right)\left(-r+s\right)}+\frac{r\left(r+s\right)}{\left(r+s\right)\left(-r+s\right)}$$

Since \frac{-r^{2}-2rs}{\left(r+s\right)\left(-r+s\right)} and \frac{r\left(r+s\right)}{\left(r+s\right)\left(-r+s\right)} have the same denominator, add them by adding their numerators.

$$\frac{-r^{2}-2rs+r\left(r+s\right)}{\left(r+s\right)\left(-r+s\right)}$$

Do the multiplications in -r^{2}-2rs+r\left(r+s\right).

$$\frac{-r^{2}-2rs+r^{2}+rs}{\left(r+s\right)\left(-r+s\right)}$$

Combine like terms in -r^{2}-2rs+r^{2}+rs.

$$\frac{-rs}{\left(r+s\right)\left(-r+s\right)}$$

Expand \left(r+s\right)\left(-r+s\right).

$$\frac{-rs}{-r^{2}+s^{2}}$$