Sure, let's determine the result of the operation [tex]\((4 + 3i) \div (1 - 2i)\)[/tex].
To solve [tex]\((4 + 3i) \div (1 - 2i)\)[/tex], we will perform the following steps:
1. Multiply the numerator and the denominator by the conjugate of the denominator:
The conjugate of [tex]\( (1 - 2i) \)[/tex] is [tex]\( (1 + 2i) \)[/tex].
[tex]\[
\frac{4 + 3i}{1 - 2i} \times \frac{1 + 2i}{1 + 2i}
\][/tex]
2. Distribute (expand) both the numerator and the denominator:
The numerator:
[tex]\[
(4 + 3i) \times (1 + 2i) = 4 \times 1 + 4 \times 2i + 3i \times 1 + 3i \times 2i
\][/tex]
Simplifying, we get:
[tex]\[
= 4 + 8i + 3i + 6i^2
\][/tex]
Remember that [tex]\( i^2 = -1 \)[/tex], so:
[tex]\[
= 4 + 11i + 6(-1) = 4 + 11i - 6 = -2 + 11i
\][/tex]
The denominator:
[tex]\[
(1 - 2i) \times (1 + 2i) = 1 \times 1 + 1 \times 2i - 2i \times 1 - 2i \times 2i
\][/tex]
Simplifying, we get:
[tex]\[
= 1 + 2i - 2i - 4i^2
\][/tex]
Remember that [tex]\( i^2 = -1 \)[/tex], so:
[tex]\[
= 1 - 4(-1) = 1 + 4 = 5
\][/tex]
3. Combine the results:
[tex]\[
\frac{-2 + 11i}{5}
\][/tex]
4. Separate the real and imaginary parts by dividing each part by the denominator:
Real part:
[tex]\[
\frac{-2}{5} = -0.4
\][/tex]
Imaginary part:
[tex]\[
\frac{11i}{5} = 2.2i
\][/tex]
Thus, the quotient is:
[tex]\[
-0.4 + 2.2i
\][/tex]
So, the result of [tex]\( \frac{4 + 3i}{1 - 2i} \)[/tex] is [tex]\( -0.4 + 2.2i \)[/tex].
