+44 How do you find values of a and b, if you are given a+b and ab (and also a^3 + b^3)?
The question I have is: find all the possible solutions of a and b, given a+b=14 and a^3+b^3=812.
Using this information, I find ab, but I'm not sure how I can use it to find the actual answer.
Note that \begin{align*} a^3 + b^3 &= (a + b)(a^2 - ab + b^2) \ &= 14(a^2 - ab + b^2). \end{align*}Since a2−ab+b2=(a−b)2+2ab, we can factor the left-hand side as follows: [a^3 + b^3 = 14(a - b)^2 + 28ab = 2(7(a - b)^2 + 14ab).]Since a+b=14, a−b must be an integer. Thus, the possible values of (a−b)2 are 1, 4, 9, and 16. The possible values of a and b are given as follows:
\begin{tabular}{c|c|c} (a−b)2 & a & b \ \hline 1 & 7 & 7 \ 4 & 8 & 6 \ 9 & 10 & 4 \ 16 & 11 & 3 \end{tabular}
Therefore, the possible solutions are (a,b)=(7,7), (8,6), (10,4), and (11,3).
