Question

a + b = 2
ab = 4
a^2 + b^2 = -4

a^3= ..?

Answer 1

taking first two equations and combining them:
\(a=(2-b)\) substituting in 2º,
\((2-b)b=4\)
\(-b^2+2b-4=0\)
solutions are :\(b_1=1+i\sqrt{3}\) and \(b_2=1-i\sqrt{3}\)
now: \(a_1= 2-1-i\sqrt{3}=1-i\sqrt3 \) and \(a_2=2-1+i\sqrt{3}=1+i\sqrt3 \)
to get a^3, just solve this:
\((1-i\sqrt3)^3=?\)

Answer 2

ahh, fogot to mention. The intuition that there is no real solution, is in a^2 + b^2 = -4
.