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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.

Jul 11, 2023

#1
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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).

Jul 11, 2023
#2
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a+b=14,

a^3+b^3=812, solve for a, b

use substitution to get:

a = 7 - sqrt(3)    and     b = 7 + sqrt(3)

a = 7 + sqrt(3)   and     b = 7 - sqrt(3)

Jul 13, 2023