SOLUTION: Evaluate √(-5+12i) (or) (-5+12i)½ ?

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: Evaluate √(-5+12i) (or) (-5+12i)½ ?      Log On


   

Question 1098569: Evaluate √(-5+12i) (or) (-5+12i)½ ?
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Let z=a%2Bbi, so then
z%5E2=%28a%2Bbi%29%28a%2Bbi%29=a%5E2-b%5E2%2B%282ab%29i
So then in this case,
1.a%5E2-b%5E2=-5
2.2ab=12
From 2,
ab=6
a=6%2Fb
a%5E2=36%2Fb%5E2
Substituting into 1,
36%2Fb%5E2-b%5E2=-5
36-b%5E4=-5b%5E2
b%5E4-5b%5E2-36=0
Use a substitution,
u=b%5E2
u%5E2-5u-36=0
%28u-9%29%28u%2B4%29=0
So there are two u solutions,
u-9=0
u=9
Since we assume that a and b are real numbers we only solve for this solution.
The other would lead to an imaginary number.
So then when u=9,
b%5E2=9
b=0+%2B-+3
When b=0+%2B-3+,
a%280+%2B-+3%29=6
a=0+%2B-+2,
From eq. 2, the product is positive so a and b have the same sign.
z=2%2B3i%29 and z=-%282%2B3i%29