2015/108) If $a + b = 1$ and $a^2 + b^2 = 2$ what is the value of $a^3 + b^3$ ?

we have

$a+b =1 \cdots(1)$
$a^2+b^2 = 2 \cdots(2)$
from 1st we get
$(a+b)^2 = a^2 +b^2 + 2ab = 2 + 2ab = 1$
or $ab = \dfrac{- 1}{2}$

hence
$a^3+b^3 = (a+b)^3 - 3ab(a+b) = 1 - 3 * \dfrac{-1}{2} * 1 = \dfrac{5}{2}$