# What are the real part and the imaginary part of complex number z if z=square root(2+6i) ?

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll write the rectangular form of a complex number:

z = a + bi

a = the real part = Re(z)

b = the imaginary part = Im(z)

We'll raise to square both sides:

z^2 = (a+bi)^2

z^2 = a^2 + 2abi + b^2*i^2, but i^2  =-1

z^2 = a^2 + 2abi - b^2

But z^2 = 2 + 6i

Comparing, we'll get:

a^2 + 2abi - b^2 = 2 + 6i

a^2 - b^2 = 2 (1)

2ab = 6

ab = 3

b = 3/a (2)

We'll substitute (2) in (1):

a^2 - 9/a^2 = 2

We'll multiply by a^2 all over:

a^4 - 2a^2 - 9 = 0

We'll substitute a^2 = t

t^2 - 2t - 9 = 0

t1 = [2 + sqrt(4 + 36)]/2

t1 = (2+sqrt40)/2

t1 = 1+sqrt10

a^2 = 1+sqrt10

a1 = +sqrt (1+sqrt10) and a2 = -sqrt (1+sqrt10)

b1 = 3/a1 = 3/sqrt (1+sqrt10)

b1 = 3*sqrt (1+sqrt10)/(1+sqrt10)

b2 = -3*sqrt (1+sqrt10)/(1+sqrt10)

The real and imaginary parts of z are: Re(z) = {-sqrt (1+sqrt10) sqrt (1+sqrt10)}  and Im(z) = {-3*sqrt (1+sqrt10)/(1+sqrt10) ; 3*sqrt (1+sqrt10)/(1+sqrt10)}.