SOLUTION: Solve the following for z: {{{ z^2 = 5 - 12i }}}

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: Solve the following for z: {{{ z^2 = 5 - 12i }}}      Log On


   

Question 1178611: Solve the following for z: +z%5E2+=+5+-+12i+
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Solve the following for z:
z%5E2=5-12i
substitute z=x%2Byi
%28x%2Byi%29%5E2=5-12i................expand
x%5E2%2B2yi%2B%28yi%29%5E2=5-12i
x%5E2%2B2yi%2By%5E2%28i%29%5E2=5-12i
x%5E2%2B2yi%2By%5E2%28-1%29=5-12i
x%5E2%2B2yi-y%5E2=5-12i
%28x%5E2-y%5E2%29+%2B2yi=5-12i
complex numbers can be equal only if their real and imaginary parts are equal
so,
x%5E2-y%5E2=5=>%28x+-+y%29+%28x+%2B+y%29+=+5: integer solutions that satisfy this are: x = ± 3, y = ±+2
2yi=-12i=> yi=-6i
substitute back z=x%2Byi
if x=3 and y=2
z=3%2B2i-> also z=3-2i
or
if x=-3 and y=2
z+=+-3+%2B+2+i-> also z+=+-3+-+2+i