Question

If z is a non-real complex number and z^2−26z is a real number, what is the value of z+z¯?


z¯ denotes the conjugate of the complex number z.

Answer 1

Let z=a+bi where i²=-1
then
z²-26z
=-b²-26a + (2a-26)bi
Since is real, and z is non-real,
therefore b≠0, and consequently
(2a-26)bi = 0
=>
2a-26=0,
=>
a=13

Thus z=13+bi, z(conjugate) = 13-bi
so
z+z(conjugate)=13+bi+13-bi=26