help with a math problem plz.?

Question

suppose that the sum of the squares of two complex numbers x and y is 7 and the sum of their cubes is 10. What is the largest real value that x+y can have?

thx.

plz show work. thx

Answer 1

x = a+bi
y = c+di
x^2 = a^2 +2abi -b^2
y^2 = c^2 +2cdi -d^2
x^2+y^2 =a^2-b^2 +c^2-d^2 +2i(ab+cd) = 7
For this to be true we need:
ab = -cd <-- Eq 1 and,
a^2 -b^+c^2-d^2 = 7 <-- Eq 2

Now do the same with x^3 + y^3 = 10 and get two more equations in a,b,c,d.

Solve these 4 equations simultaneosly and you have your solution.