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140 4. Equations

(a) z2 + 2% + 3 = 0

(b) .z2 + 22 - 3 = 0

(cl z2 -6z+9=0

(d) z2 + 2iz - 1 = 0

(e) z2 + (1 + 2i)z + (-1 + i) = 0.

6. (a) Let p and q be nonzero reals. Show that the cubic equation

z3+pz+q=o

is equivalent through the transformation z = x+yi to the system

x3-3xy2+px+qrO (4

y(3z2 - y” + p) = 0. m

(b) Show that the locus of (B) consists of the x-axis along with a

hyperbola whose center is

lines y = &z and y = - $

0,O) and whose asymptotes are the

3x. How does the sign of p determine

which of the x-axis and the y-axis is the transverse axis of the

hyperbola?

(c) Verify that the equation (A) can be written in the form

y2 = (1/3)(x2 + P + n/x).

(d) Verify that

Jx2 + P + (!7/x> - x = [p + (Q/X)]/ [ Jx2 + P + w4 + x]^7

and deduce that the locus of (A) is asymptotic to the pair of

straight lines whose equation is 3y2 = x2.

(e) Show that (A) is asymptotic to the y-axis.

(f) Sketch the graphs of (A) and (B) on the same axes, indicating

where the curves are likely to intersect. Then draw a circle with

center at the origin whose radius is large enough that all the

points of intersection of (A) and (B) are in its interior. Label

all the intersection points of (A) with this circle by the letter

R, and all the intersection points of (B) with this circle by the

letter I. Verify that the letters R and I alternate.

7. Carry out the procedure of Exercise 6 on the equation z3 + z + 1 = 0.
   Show that the equivalent real system is

x3-3xy2+2+1=0 y(3x2 - y2 + 1) = 0.

Sketch the graphs of these two curves, and verify that the cubic equa-

tion has one real solution between -1 and 0 and two nonreal complex

conjugate solutions.