A and B are complex numbers such that:

A² + B² = 7
A³ + B³ = 10

What is the maximum possible real value of A + B?

Changing variables to u and v where u = A + B and v = A – B,

i.e. substituting A = (u + v)/2 and  B = (u – v)/2 :

    (u + v)²/4 + (u – v)²/4 = 7   which gives        u² + v² = 14  (1)

& (u + v)³/8 + (u – v)³/8 = 10  which gives  u(u² + 3v²) = 40  (2)

From (1), v² = 14 – u², which can now be substituted into (2):

            u(u² + 42 – 3u²) = 40

                 u³ -21u + 20 = 0

     (u – 1)(u – 4)(u + 5) = 0  giving         a + b = u = 1, 4, -5

Was I alone in wondering whether ‘maximum real value of a + b’
was meant to be ‘maximum real part of a + b’? All is well, it turns
out that a + b can only have real values, and the maximum is 4.

Posted by Harry on 2015-12-12 15:48:36