+130077 a + b = 14
Square both sides
a^2 + 2ab + b^2 = 196 ⇒ a^2 + b^2 = 196 - 2ab
a^3 + b^3 = (a + b) ( a^2 - ab + b^2)
So
(a + b) (a^2 - ab + b^2) = 812 + 4ab
(14) ( 196 - 2ab - ab) = 812 + 4ab
14 ( 196 - 3ab) = 812 + 4ab
2744 - 42ab = 812 + 4ab
1932 = 46ab
42 = ab
42/a = b
a + b =14
a + 42/a = 14
a^2 + 42 = 14a
a^2 - 14a = -42
a^2 - 14a + 49 = -42 + 29
(a - 7)^2 = 7 take both roots
a - 7 = ± sqrt 7
a = 7 + sqrt 7 and b = 7 - sqrt 7
or
a = 7 - sqrt 7 and b = 7 + sqrt 7
