How to solve this homogeneous recurrence relation of 2nd order? I have this homogeneous recurrence relation: $x_n = 3x_{n-1} + 2x_{n-2}$ for $n \geq 2$ and $x_0 = 0$, $x_1 = 1$. I form the characteristic polynomial: $r^2 - 3r -2 = 0$ which gives the roots $r = \frac{3}{2} - \frac{\sqrt{17}}{2}$, $\frac{3}{2} + \frac{\sqrt{17}}{2}$. Now when I pluck these values into the function $a_n = \alpha (\frac{3}{2} - \frac{\sqrt{17}}{2})^n + \beta (\frac{3}{2} + \frac{\sqrt{17}}{2})^n$ and solve it for $\alpha$ and $\beta$ I get enormous fractions which make my head hurt. Is there an easier and neater way of solving this or am I doing something wrong? 

Name the roots $r$ and $s$ and solve for the initial conditions
$$ar^0+bs^0=0=a+b,\\ar^1+bs^1=1=ar+bs.$$
Then
$$a=-b=\frac1{r-s}.$$
The difference between the roots being $\sqrt{17}$, the "enormous fraction" is
$$x_n=\frac{\left(\frac{3+\sqrt{17}}2\right)^n-\left(\frac{3-\sqrt{17}}2\right)^n}{\sqrt{17}}=\frac{(3+\sqrt{17})^n-(3-\sqrt{17})^n}{2^n\sqrt{17}}.$$
