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Find all solutions to the system
\(\begin{align*} a + b &= 14, \\ a^3 + b^3 &= 812. \end{align*}\)

 Oct 1, 2022
 #1
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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.

 Oct 2, 2022

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