Finding $x^3+y^3$, given $x + y = 5$ and $xy = 1$ 
Given
$$x + y = 5 \qquad xy = 1$$
Find $x^3 + y^3$.

To solve this, I tried this:
$y = \frac{1} {x}$
$x + \frac{1}{x} = 5$
$x^2 - 5x + 1 = 0$
What formula needs to be used to find the value of $x$?

Note that $x^3+y^3=(x+y)(x^2-xy+y^3)$. We have $x+y=5$. Then $(x+y)^2=x^2+y^2+2xy=25$. Then, $x^2-xy+y^2=25-3xy=22$. All that is left is to multiply them.
