+219 Let $a$ and $b$ be complex numbers. If $a + b = 4$ and $a^2 + b^2 = 6 + 2ab,$ then what is $a^3 + b^3?$
+1926 With the first equation, let's square both sides. We get
\(a^2 + 2ab + b^2 = 16 \\ a^2 + b^2 = 16 - 2ab \)
Now, we have a second equation of
\(a^2 + b^2 = 6 + 2ab \)
Now, we simplfy subtract the first equation from the second one, we get
\(0 = -10 + 4ab \\ 10 = 4ab \\ ab = 10 / 4 = 5/2 \\ a^3 + b^3 = (a + b) ( a^2 + b^2 - ab) \\ a^3 + b^3 = (4) ( 6 + 2ab - ab) \\ a^3 + b^3 = (4) ( 6 - ab) \\ a^3 + b^3 = (4) ( 6 - 5/2) \\ a^3 + b^3 = (4) ( 7/2) \\ a^3 + b^3 = 14\)
14 is our answer.
Thanks! :)
