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0
14
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avatar+1420 

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

 Mar 22, 2024
 #1
avatar+129732 
+1

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

 

 

cool cool cool

 Mar 22, 2024

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