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Find all solutions to the system
a + b = 14
a^3 + b^3 = 812 + a^2 + b^2

 Jan 3, 2023
 #1
avatar+1317 
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Remember: (You can simplify by expanding out)

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

and

a^2 + b^2 = (a + b)^2 - 2ab

Substituting in, you get:

(a + b)(a^2 - ab + b^2) = 812 + (a + b)^2 - 2ab

14[(a + b)^2 - 2ab - ab] = 812 + 196 - 2ab

14(196 - 3ab) = 1008 - 2ab 

2744 - 42ab = 1008 - 2ab

1736 = 40ab

ab = 43.4

a + b = 14

Write a quadratic equation using vieta's formula:

x^2 - 14x + 43.4 = 0. x is the values of a and b.

a and b are 14 +- sqrt(22.4)) / 2

 Jan 3, 2023

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