Question

$a+b=3\\a^3+b^3=15\\a^2 + b^2=?$

Answer 1

I.

$a+b=3 => (a+b)^{2}=3^{2} => a^{2}+2ab+b^{2}=9 => a^{2}+b^{2}=9-2ab$

II.

$a^{3}+b^{3}=15 => (a+b)(a^{2}-ab+b^{2})=15 => 3(9-2ab-ab)=15 =>-3ab = -4 <=> ab=\frac{4}{3}$

Substituindo II em I:

$a^{2}+b^{2} = 9-2ab => a^{2}+b^{2} = 9-2*\frac{4}{3} = 9-\frac{8}{3} = \frac{19}{3}$