<s>[INST] Finding $a$ and $b$ from $a^3+b^3$ and $a^2+b^2$ Question 1 
Two numbers are such that the sum of their cubes is 14 and the sum of their squares is 6. Find the sum of the two numbers.
I did 
$a^2+b^2=6$ and $a^3+b^3=14$ Find $a$ and $b$, two numbers. but got lost when trying to algebraicly solve it.
Thank you, Any help is appreciated
 [/INST] The problem doesn't specify any restriction on the numbers $a$ and $b$, so we're going to assume they are complex.  Let $S=a+b$ denote the sum.  Then
$$S^2=(a+b)^2=a^2+2ab+b^2=6+2ab$$
and
$$S^3=(a+b)^3=a^3+3a^2b+3ab^2+b^3=14+3abS$$
Rewriting the first of these as $2ab=S^2-6$ and multiplying both sides of the second by $2$ gives
$$2S^3=28+3(2ab)S=28+3(S^2-6)S=28+3S^3-18S$$
or
$$S^3-18S+28=(S-2)(S^2+2S-14)=0$$
The possible values of $S$ are thus $2$, $-1+\sqrt{15}$, and $-1-\sqrt{15}$.  
If you care to chase down the actual values of $a$ and $b$, the first two of these give real values while the third gives a pair of complex conjugates.  In particular, $S=2$ leads to $1\pm\sqrt2$ for $a$ and $b$.
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