Solving for a and b: Fast Method for a^4 + b^4 Calculation

[Taturana]

Homework Statement

Given that a-b = 5 and ab = 2, what is the value of a^4 + b^4?

Homework Equations

The Attempt at a Solution

Doing the math I know that
$$a_{1} = \frac{5+\sqrt{33}}{2}$$
$$a_{2} = \frac{5-\sqrt{33}}{2}$$

$$b_{1} = \frac{-5+\sqrt{33}}{2}$$
$$b_{2} = \frac{-5-\sqrt{33}}{2}$$

So my question is: there is any fast way to do $$a^{4} + b^{4}$$?

 

[phyzguy]

You can expand a^4+b^4 so you never have to solve for a and b, as follows:
$$a^4+b^4=(a-b)^4+4a^3b-6a^2b^2+4ab^3=(a-b)^4+4ab(a^2+b^2)-6a^2b^2=(a-b)^4-6(ab)^2+4ab((a-b)^2+2ab) = (a-b)^4+2(ab)^2+4ab(a-b)^2$$

Now you just plug in what you know for (a-b) and ab.