+0  
 
-2
1
1
avatar+195 

Let $a$ and $b$ be complex numbers. If $a + b = 4$ and $a^2 + b^2 = 6 + 2ab,$ then what is $a^3 + b^3?$

 Jun 5, 2024
 #1
avatar+806 
+1

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! :)

 Jun 5, 2024

0 Online Users