Express 2 (2-i)(2+I)-1 in the form of a+ib

We need to simplify the expression \( 2(2 - i)(2 + i) - 1 \) and express the result in the form \( a + ib \), where \( a \) and \( b \) are real numbers.

### Step 1: Simplify \( (2 - i)(2 + i) \)
Notice that \( (2 - i)(2 + i) \) is a difference of squares. We can apply the formula:
\[
(a - b)(a + b) = a^2 - b^2
\]
Here, \( a = 2 \) and \( b = i \), so:
\[
(2 - i)(2 + i) = 2^2 - i^2 = 4 - (-1) = 4 + 1 = 5
\]

### Step 2: Multiply by 2
Now we multiply the result by 2:
\[
2(2 - i)(2 + i) = 2 \times 5 = 10
\]

### Step 3: Subtract 1
Finally, subtract 1 from the result:
\[
2(2 - i)(2 + i) - 1 = 10 - 1 = 9
\]

### Final Answer:
The expression simplifies to \( 9 + 0i \), which means the real part is 9 and the imaginary part is 0. Therefore, the expression in the form \( a + ib \) is:
\[
9 + 0i
\]

Would you like more details or have any questions?

Here are five related questions to deepen your understanding:
1. What is the difference of squares, and why does it apply here?
2. How does the value of \( i^2 \) influence calculations involving complex numbers?
3. Can you verify that \( (2 - i)(2 + i) = 5 \) by expanding directly?
4. How would this calculation change if the imaginary unit \( i \) was replaced by another constant?
5. What are some other common operations involving complex numbers?

**Tip:** Remember, multiplying a complex number by its conjugate always results in a real number!

Express 2(2-i)(2+i)-1 in the form of a+ib

Learn how to express the complex number expression 2(2-i)(2+i)-1 in the form a + ib. This solution simplifies the difference of squares and provides a step-by-step breakdown suitable for high school students studying complex numbers and algebraic operations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation