Math Problem Statement
Express 2 (2-i)(2+I)-1 in the form of a+ib
Solution
We need to simplify the expression and express the result in the form , where and are real numbers.
Step 1: Simplify
Notice that is a difference of squares. We can apply the formula: Here, and , so:
Step 2: Multiply by 2
Now we multiply the result by 2:
Step 3: Subtract 1
Finally, subtract 1 from the result:
Final Answer:
The expression simplifies to , which means the real part is 9 and the imaginary part is 0. Therefore, the expression in the form is:
Would you like more details or have any questions?
Here are five related questions to deepen your understanding:
- What is the difference of squares, and why does it apply here?
- How does the value of influence calculations involving complex numbers?
- Can you verify that by expanding directly?
- How would this calculation change if the imaginary unit was replaced by another constant?
- What are some other common operations involving complex numbers?
Tip: Remember, multiplying a complex number by its conjugate always results in a real number!
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Math Problem Analysis
Mathematical Concepts
Complex Numbers
Algebra
Difference of Squares
Formulas
(a - b)(a + b) = a^2 - b^2
Theorems
Difference of Squares
Suitable Grade Level
Grades 9-12