+23
Find all solutions to the system
\(\begin{align*} a + b &= 14, \\ a^3 + b^3 &= 812. \end{align*}\)
+1622 Sum of cubes => a^3 + b^3 = (a + b)(a^2 - ab + b^2)
a^2 - ab + b^2 = (a + b)^2 - 2ab - ab
a^3 + b^3 = 14((14)^2 - 3ab)
812/14 = 196 - 3ab
58 - 196 = -3ab
ab = 46
a + b = 14.
By vieta's formula, we can write a quadratic: x^2 - 14x + 46 = 0, where a and b are the solutions to the quadratic.
[14 +- sqrt(196 - 184)]/2
(14 +- sqrt(12))/2
(14 +- 2sqrt(3))/2
7 +- sqrt(3).
Thus (a, b) = \((7 + \sqrt{3}, 7 - \sqrt{3}) \) and you can reverse a and b in the ordered pair.
