Find all solutions to the system
a + b = 14
a^3 + b^3 = 812.
Please help! I don't understand how to do this.
+128448 a + b =14 (1)
Square both sides of this
a^2 + 2ab + b^2 = 196 rearrange as
a^2 + b^2 =196 - 2ab (2)
a^3 + b^3 = 812
Using the formula for the sum of cubes we have that
(a + b) ( a^2 - ab + b^2) = 812
sub in (1) and (2)
(14) ( 196 - 2ab - ab) = 812 divide both sides by 14
(196 - 3ab) = 58 rearrange as
3ab = 196 - 58
3ab = 138 divide both sides by 3
ab = 46
b = 46 / a
sub this into 1
a + 46/a = 14 multiply through by a
a^2 + 46 = 14a rearrange as
a^2 - 14a + 46 = 0 complete the square
a^2 - 14a + 49 = -46 + 49
(a - 7)^2 = 3 take both roots
a - 7 = ±sqrt (3)
a = 7 + sqrt (3) or a = 7 - sqrt (3)
And b = the conjugate of either answer
So ( a, b) = ( 7 + sqrt (3), 7 - sqrt (3) ) or ( 7 - sqrt (3) , 7 + sqrt (3))
