Certainly! Let's solve the expression [tex]\((2 - i \sqrt{3})^2\)[/tex] step-by-step, where [tex]\(i\)[/tex] represents the imaginary unit.
1. Expression Setup:
We start with [tex]\((2 - i \sqrt{3})^2\)[/tex].
2. Complex Number Squaring Formula:
We will apply the formula for squaring a complex number. For a complex number [tex]\(a + bi\)[/tex], the squared form is given by:
[tex]\[
(a + bi)^2 = a^2 + 2abi + (bi)^2
\][/tex]
3. Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
In our case:
[tex]\[
a = 2, \quad b = -\sqrt{3}
\][/tex]
Since the imaginary part is [tex]\(-i\sqrt{3}\)[/tex], [tex]\(b\)[/tex] will be [tex]\(-\sqrt{3}\)[/tex].
4. Square the real part:
[tex]\[
a^2 = 2^2 = 4
\][/tex]
5. Square the imaginary part:
[tex]\[
(bi)^2 = (-\sqrt{3}i)^2 = (\sqrt{3})^2 \cdot (i)^2 = 3 \cdot (-1) = -3
\][/tex]
6. Determine the middle term:
[tex]\[
2abi = 2 \cdot 2 \cdot (-\sqrt{3}) \cdot i = -4\sqrt{3}i
\][/tex]
7. Combine all parts:
Putting it all together, we get:
[tex]\[
a^2 + 2abi + (bi)^2 = 4 + (-4\sqrt{3}i) - 3
\][/tex]
8. Combine real parts and imaginary parts:
[tex]\[
4 - 3 + (-4\sqrt{3}i) = 1 - 4\sqrt{3}i
\][/tex]
Therefore, the result of squaring the complex number [tex]\((2 - i \sqrt{3})^2\)[/tex] is:
[tex]\[
1 - 4\sqrt{3}i
\][/tex]
In terms of real and imaginary parts, we can express this result as:
- Real part: [tex]\(1\)[/tex]
- Imaginary part: [tex]\(-4\sqrt{3}\)[/tex]
Hence, the final answer to the expression [tex]\((2 - i \sqrt{3})^2\)[/tex] is:
[tex]\[
1 - 6.928203230275509i
\][/tex]
