To determine which of the given options is the square root of the complex number [tex]\(8 - 6i\)[/tex], we proceed as follows:
Given the complex number [tex]\(8 - 6i\)[/tex], we are looking for a complex number [tex]\(z\)[/tex] such that when squared, it equals [tex]\(8 - 6i\)[/tex]. Let's denote [tex]\(z\)[/tex] as [tex]\(a + bi\)[/tex] where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are real numbers, and [tex]\(i\)[/tex] is the imaginary unit with [tex]\(i^2 = -1\)[/tex].
### Step-by-Step Solution:
1. Express [tex]\(z\)[/tex] and its square:
[tex]\[
z = a + bi
\][/tex]
[tex]\[
z^2 = (a + bi)^2 = a^2 + 2abi + (bi)^2 = a^2 + 2abi - b^2 = (a^2 - b^2) + 2abi
\][/tex]
2. Equate the square to [tex]\(8 - 6i\)[/tex]:
[tex]\[
(a^2 - b^2) + 2abi = 8 - 6i
\][/tex]
3. Match real and imaginary parts:
From the equation, we separate the real and imaginary parts:
[tex]\[
a^2 - b^2 = 8 \quad \text{(Real part)}
\][/tex]
[tex]\[
2ab = -6 \quad \text{(Imaginary part)}
\][/tex]
4. Solve the imaginary part equation for one of the variables:
[tex]\[
2ab = -6
\][/tex]
[tex]\[
ab = -3
\][/tex]
5. Check different possible values for [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
- For [tex]\(a = 3\)[/tex], solve for [tex]\(b\)[/tex]:
[tex]\[
3b = -3
\][/tex]
[tex]\[
b = -1
\][/tex]
Substitute [tex]\(a = 3\)[/tex] and [tex]\(b = -1\)[/tex] into the real part equation:
[tex]\[
a^2 - b^2 = 3^2 - (-1)^2 = 9 - 1 = 8
\][/tex]
This holds true.
So, one possible solution is:
[tex]\[
z = 3 - i
\][/tex]
Given the options:
- F: [tex]\(3 + i\)[/tex]
- G: [tex]\(3 - i\)[/tex]
- H: [tex]\(3 - 4i\)[/tex]
- J: [tex]\(4 + 3i\)[/tex]
- K: [tex]\(4 - 3i\)[/tex]
Among these options, the correct one that matches our solution is:
[tex]\[
\boxed{G \ \ (3 - i)}
\][/tex]
So, the square root of [tex]\(8 - 6i\)[/tex] is [tex]\(3 - i\)[/tex]. Therefore, the answer is G.
