+1839 Let $a$ and $b$ be complex numbers. If $a + b = 4$ and $a^2 + b^2 = 6 + 2ab,$ then what is $a^3 + b^3?$
+129852 a + b = 4 square both sides
a^2 + 2ab + b^2 = 16
a^2 + b^2 = 16 - 2ab (1)
And we are given that
a^2 + b^2 = 6 + 2ab (2)
Subtract (1) from ( 2)
0 = -10 + 4ab
10 = 4ab
ab = 10 / 4 = 5/2
a^3 + b^3 = (a + b) ( a^2 + b^2 - ab) =
(4) ( 6 + 2ab - ab) =
(4) ( 6 - ab) =
(4) ( 6 - 5/2) =
(4) ( 7/2) =
14
