https://r2.erweima.ai/mathbot/image/1718214276108mii0vdrl.png

To find the quotient of the complex numbers \(\frac{3 + 2i}{1 - 5i}\), we need to multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(1 - 5i\) is \(1 + 5i\).

\[
\frac{3 + 2i}{1 - 5i} \cdot \frac{1 + 5i}{1 + 5i}
\]

First, we compute the numerator:

\[
(3 + 2i)(1 + 5i) = 3 \cdot 1 + 3 \cdot 5i + 2i \cdot 1 + 2i \cdot 5i = 3 + 15i + 2i + 10i^2
\]

Since \(i^2 = -1\),

\[
10i^2 = 10(-1) = -10
\]

So, the numerator becomes:

\[
3 + 15i + 2i - 10 = -7 + 17i
\]

Next, we compute the denominator:

\[
(1 - 5i)(1 + 5i) = 1 \cdot 1 + 1 \cdot 5i - 5i \cdot 1 - 5i \cdot 5i = 1 + 5i - 5i - 25i^2
\]

Again, since \(i^2 = -1\),

\[
-25i^2 = -25(-1) = 25
\]

So, the denominator becomes:

\[
1 + 25 = 26
\]

Therefore, the quotient is:

\[
\frac{-7 + 17i}{26} = -\frac{7}{26} + \frac{17i}{26}
\]

Which simplifies to:

\[
-\frac{7}{26} + \frac{17i}{26}
\]

This matches option B. So, the answer is:

\[
\boxed{\text{B. } -\frac{7}{26} + \frac{17i}{26}i}
\]

Do you want details or have any questions? Here are five more questions you might ask:

1. How do you find the conjugate of a complex number?
2. What is the significance of multiplying by the conjugate in complex number division?
3. How do you simplify a complex fraction?
4. What are the properties of imaginary numbers?
5. How do you multiply two complex numbers?

**Tip:** Always express the final result of a complex number in the standard form \(a + bi\), where \(a\) and \(b\) are real numbers.

Learn how to solve the complex number division problem of (3 + 2i)/(1 - 5i) step-by-step. This problem involves operations with complex numbers and applies the complex conjugate formula and complex number division formula. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation