To divide the complex numbers [tex]\(\frac{-20 + 40i}{-6 - 2i}\)[/tex], we can simplify the expression by following these steps:
1. Identify the given complex numbers:
The numerator is [tex]\(-20 + 40i\)[/tex] and the denominator is [tex]\(-6 - 2i\)[/tex].
2. Multiply the numerator and the denominator by the conjugate of the denominator:
The conjugate of [tex]\(-6 - 2i\)[/tex] is [tex]\(-6 + 2i\)[/tex]. We multiply both the numerator and the denominator by this conjugate to eliminate the imaginary part from the denominator:
[tex]\[
\frac{(-20 + 40i)(-6 + 2i)}{(-6 - 2i)(-6 + 2i)}
\][/tex]
3. Expand the numerator:
[tex]\[
(-20 + 40i)(-6 + 2i) = (-20) \cdot (-6) + (-20) \cdot (2i) + (40i) \cdot (-6) + (40i) \cdot (2i)
\][/tex]
Simplifying each term:
[tex]\[
= 120 - 40i - 240i - 80i^2
\][/tex]
Since [tex]\(i^2 = -1\)[/tex]:
[tex]\[
= 120 - 40i - 240i + 80
\][/tex]
Combine like terms (real parts and imaginary parts separately):
[tex]\[
= (120 + 80) + (-40i - 240i)
\][/tex]
[tex]\[
= 200 - 280i
\][/tex]
4. Expand the denominator:
[tex]\[
(-6 - 2i)(-6 + 2i) = (-6)^2 - (2i)^2
\][/tex]
Since [tex]\((-6)^2 = 36\)[/tex] and [tex]\((2i)^2 = 4i^2 = 4(-1) = -4\)[/tex]:
[tex]\[
= 36 - (-4)
\][/tex]
[tex]\[
= 36 + 4
\][/tex]
[tex]\[
= 40
\][/tex]
5. Divide the simplified numerator by the simplified denominator:
[tex]\[
\frac{200 - 280i}{40}
\][/tex]
Separate into real and imaginary parts:
[tex]\[
= \frac{200}{40} - \frac{280i}{40}
\][/tex]
Simplify each part:
[tex]\[
= 5 - 7i
\][/tex]
Therefore, the simplified form of the complex division [tex]\(\frac{-20 + 40i}{-6 - 2i}\)[/tex] is:
[tex]\[
1.0000000000000002 - 6.999999999999999i
\][/tex]
The real part is [tex]\(1.0000000000000002\)[/tex] and the imaginary part is [tex]\(-6.999999999999999\)[/tex]. Therefore, the result as a complex number is:
[tex]\[
1.0000000000000002 - 6.999999999999999i
\][/tex]
