<s>[INST] Let $a$ and $b$ be real numbers satisfying $a^3 - 3ab^2 = 47$ and $b^3 - 3a^2 b = 52$. Find $a^2 + b^2$. Let $a$ and $b$ be real numbers satisfying $a^3 - 3ab^2 = 47$ and $b^3 - 3a^2 b = 52$. Find $a^2 + b^2$.
I observed that adding the two has a suspicious similarity to (a+b)^3, but I cannot relate the two.  Can someone help me?
Any help is appreciated!
 [/INST] Good try with $(a+b)^3$, but you will find it better to expand
$$\eqalign{(a^2+b^2)^3
  &=a^6+3a^4b^2+3a^2b^4+b^6\cr
  &=(a^3-3ab^2)^2+(b^3-3a^2b)^2\ .\cr}$$
If you have begun studying complex numbers, you can use the hint from Joey Zou to explain the "miracle" factorisation in the last line.  However if you don't know about complex numbers yet, just expand the last line to check that it is correct.
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