You can put this solution on YOUR website! z^2=5-12i
Where z=a+bi
:
Replace z with (a+bi)
(a+bi)^2 = 5 - 12i
:
FOIL (a+bi)(a+bi)
a^2 + 2abi + b^2i^2 = 5 -12i
:
i^2 = -1 therefore
a^2 + 2abi - b^2 = 5 - 12i
a^2 + 2abi = b^2 + 5 - 12i
a^2 + 2abi + 12i = b^2 + 5
:
I don't know what else you can do with it.
z = a + bi ---> = = = (since = = = // <--- since = -1)
Now you have, from the condition, an equality of these two complex numbers:
= 5 - 12i.
It means that the real parts are equal and the imaginary parts are equal:
= 5, (1)
2ab = -12. (2)
It is the system of two equations for the two unknowns "a" and "b".
To solve it, express b = = from (2) and substitute it into (1). You will get
= 5, or
- = 5.
Multiply both sides by a^2. You will get
= 0.
Factor the left side
= 0.
The equation deploys in two independent equations
1. = 0 ---> = 9 ---> a = +/- 3 ---> b = -6/a = -/+ 2.
2. = 0 ---> = -4 ---> No real solutions.
Answer. The solutions of the original equation = 5 - 12i are these two complex numbers: = 3 - 2i or/and = -3 + 2i.
In other words, the square root of the complex number 5 - 12i in the complex domain are these two complex numbers: 3 - 2i and -3 + 2i.
