I have a question in solving a root of a complex number...
Solve: Z2 = 3 - 4i
Let
z= x + yi
Rewrite as
(x + yi)2 = 3 - 4i
Expand
x2 +2xyi + y2i2 = 3 - 4i
Simplify
x2 - y2 + 2xyi = 3 - 4i
real/imag parts
x2 - y2 = 3
and
2xy = -4
xy = -2
Therefore y = -2/x
Substitute
y = -2/x into x2 - y2 = 3
giving
x^2 - 4/x2 = 3
Multiply by x2
x^4 - 4 = 3x^2
Factor
Roots **>
x^4 - 3x^2 - 4 = 0
(x2 + 1) (x2 - 4) = 0
(x2 + 1) (x + 2) (x - 2) = 0
Given xy = -2
for x = 2 then y = -1 therefore z = 2 - i
for x = -2 then y = 1 therefore z = -2 + i
OK... the 2 solutions for z shown are indicated as correct in the text,
but these are the onnly solutions given.
What about the (x2 + 1) root found in the line marked '**>'?
How do I deal with it?
Is it not used because it does not have a real result?
Have I made a mistake or procedural error?
Thanks for you assistance,
John
(uncertain parent of secondary student)
Hi John,
Back in the third line when you write the complex number z as z= x + yi there is an assumption that x any y are real. The roots of x2 + 1 are not real and hence do not give extra solutions to your problem.
Harley