# How do you simplify (2t)/(s^2 - st) + (s+t)/(t^2 - s^2)?

Oct 20, 2015

$\frac{s - 2 t}{s \left(t - s\right)}$

#### Explanation:

Your starting expression looks like this

$\frac{2 t}{{s}^{2} - s t} + \frac{s + t}{{t}^{2} - {s}^{2}}$

Notice that you can write

${s}^{2} - s t = - \left(s t - {s}^{2}\right)$

The expression becomes

$- \frac{2 t}{s t - {s}^{2}} + \frac{s + t}{{t}^{2} - {s}^{2}}$

Now focus on factoring the two denominators. For $s t - {s}^{2}$, you can use $s$ as a common factor to write

$s t - {s}^{2} = s \cdot \left(t - s\right)$

Notice that the second denominator is a difference of squares, which means that you can factor it using

$\textcolor{b l u e}{{a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)}$

In your case, you would have

${t}^{2} - {s}^{2} = \left(t - s\right) \left(t + s\right)$

The expression can thus be written as

$- \frac{2 t}{s \left(t - s\right)} + \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{\left(t + s\right)}}}}{\left(t - s\right) \textcolor{red}{\cancel{\textcolor{b l a c k}{\left(t + s\right)}}}}$

$- \frac{2 t}{s \left(t - s\right)} + \frac{1}{t - s}$

Multiply the second fraction by $1 = \frac{s}{s}$ to get

-(2t)/(s(t-s)) + s/(s(t-s)) = color(green)((s - 2t)/(s(t-s))