<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 are lots of ways of solving this. One is to note that $a$ and $b$ are the roots of the quadratic equation $$0=(x-a)(x-b)=x^2-(a+b)x+ab=x^2-px+q$$ Where we use $p=a+b$ and $q=ab$ for the unknown coefficients.
Then we have $a^2-pa+q=b^2-pb+q=0$ and adding the equations we obtain $$(a^2+b^2)-p(a+b)+2q=0$$ which becomes$$6-p^2+2q=0$$
We also have (after multiplying the quadratic equation by $x$ and substituting $a,b$ and adding):
$$(a^3+b^3)-p(a^2+b^2)+q(a+b)=0$$which becomes $$14-6p+pq=0=28-12p+2pq=28-12p+p(p^2-6)$$ (substituting for $2q$ from the first equation)
Which gives the cubic for $p$ $$p^3-18p+28=0$$ This can be solved using standard methods. Pretty much any method will come down to solving an equivalent cubic.

Note that $p=a+b$ is what you are asked to find.
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