Write

27 = 27 e^{2 πin}

. The three complex roots are

z = 3, z = - \frac{3}{2} + \frac{3 3}{2} i, z = - \frac{3}{2} - \frac{3 3}{2} i .

On the Argand diagram, these are three equally spaced points on the circle

∣ z ∣ = 3

0, 2 π /3, 4 π /3

. They are isolated points, rather than the whole circle.

ii

Solution

Put

z = x + i y

and

z^{*} = x - i y

. The first two squared terms are complex conjugates, so their sum is

2 [(x - 2)^{2} - (y + 1)^{2}]

. Also

z z^{*} = x^{2} + y^{2}

. The equation therefore reduces to

2 [(x - 2)^{2} - (y + 1)^{2}] + 2 (x^{2} + y^{2}) + 6 = 4 (x^{2} - 2 x - y + 3) = 0.

Thus the locus is the upward-opening parabola

y = (x - 1)^{2} + 2,

1 + 2 i

and axis

x = 1

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