<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] When numbers are so small you can proceed by attempt.
However $a^2+b^2=6$ doesn't have no integer solution: wlog you can take $a,b\geq0$.
So try: $a=0\Rightarrow b=\sqrt6$ (not valid in $\mathbb Z$).
$a=1\Rightarrow b=\sqrt5$ (not valid as above)
$a=2\Rightarrow b=\sqrt2$ (idem).
$a\geq3\Rightarrow b^2<0$.
But maybe you mean to find $a,b\in\mathbb R$.
In this case just notice that $14=a^3+b^3=x(6-y)$ and $6=a^2+b^2=x^2-2y$, where $x=a+b$ and $y=ab$. Solving the system you'll find $x=\frac{14}{6-y}$ and $y^3-9y^2+10=0$, which is solvable using the well known formulas for the 3rd degree polynomials.
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