<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] There's another way to get to answer if we take $a=c+d$ and $b=c-d$, then using binomial expansions we have
$$
a^2+b^2=2(c^2+d^2)=6\\
a^3+b^3=2c^3+6cd^2=14
$$
We have  $d^2=3-c^2$ from first equation and using this  in second equation
$$
c(9-2c^2)=7
$$
We can easily see that one of the solutions is c=1.
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