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To solve the division \(\frac{2 + i}{2 - i}\), follow these steps:

### Step 1: Multiply the numerator and denominator by the conjugate of the denominator.
The conjugate of \(2 - i\) is \(2 + i\). So, multiply both the numerator and denominator by \(2 + i\):

\[
\frac{2 + i}{2 - i} \times \frac{2 + i}{2 + i} = \frac{(2 + i)(2 + i)}{(2 - i)(2 + i)}
\]

### Step 2: Expand the numerator and the denominator.
1. Expand the numerator \((2 + i)(2 + i)\):

\[
(2 + i)(2 + i) = 2(2) + 2(i) + i(2) + i(i) = 4 + 2i + 2i + i^2 = 4 + 4i + (-1) = 3 + 4i
\]

2. Expand the denominator \((2 - i)(2 + i)\):

\[
(2 - i)(2 + i) = 2(2) + 2(i) - i(2) - i(i) = 4 + 2i - 2i - i^2 = 4 - (-1) = 4 + 1 = 5
\]

### Step 3: Write the result in the form \(a + bi\).

\[
\frac{3 + 4i}{5} = \frac{3}{5} + \frac{4}{5}i
\]

So the final answer is:

\[
\frac{2 + i}{2 - i} = \frac{3}{5} + \frac{4}{5}i
\]

This is in the form \(a + bi\), where \(a = \frac{3}{5}\) and \(b = \frac{4}{5}\).

Would you like more details on any of these steps, or do you have any questions?

Here are some related questions to further explore this concept:

1. How do you find the conjugate of a complex number?
2. Why does multiplying by the conjugate eliminate the imaginary part in the denominator?
3. What are some applications of complex numbers in engineering or physics?
4. How do you add and subtract complex numbers?
5. Can you provide an example of dividing complex numbers where both the numerator and denominator are purely imaginary?

**Tip:** When dealing with complex numbers, always remember to express the final answer in the standard form \(a + bi\), where \(a\) and \(b\) are real numbers.

Divide (2 + i) / (2 - i) and provide the answer in the form a + bi.

Learn how to divide the complex numbers (2 + i) by (2 - i) by multiplying by the conjugate. This detailed step-by-step solution covers key concepts like complex number conjugates and simplifying results to standard form a + bi.

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