Let a and b be complex numbers. If a + b = 4 and a^2 + b^2 = 6 + ab, then what is a^3 + b^3?
Let's first focus on the first equation.
Squaring both sides of the first equation, we find that
a2+2ab+b2=16
Reaaranging the second equation from the problem, we get that
a2−ab+b2=6
Now, moving on, let's note something important. We have that
a3+b3=(a+b)(a2+b2−ab)
We already have all the terms needed to solve the problem. We know every single number.
The first term parenthesis is the first equation, and the second parenthesis is the second equation.
Thus, plugging in 6 and 4, we get that
a3+b3=6∗4=24
So 24 is our answer.
Thanks! :)