https://file.aiquickdraw.com/mathbot/file/1726021358667flw3coq1.jpg

The problem shown in the image is:

\[
(5 + 2i)(5 - 2i)
\]

This expression represents the multiplication of two conjugates, and there's a useful formula for this kind of problem:

\[
(a + bi)(a - bi) = a^2 - (bi)^2
\]

Here, \(a = 5\) and \(b = 2\), so we can apply the formula:

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

Now, let's simplify each part:

\[
5^2 = 25
\]

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

Therefore, the expression becomes:

\[
25 - (-4) = 25 + 4 = 29
\]

So, the result of multiplying \((5 + 2i)(5 - 2i)\) is \(29\).

Would you like more details or have any questions? Here are 5 related questions:

1. What are complex conjugates and why are they useful in simplifying expressions?
2. How would you multiply complex numbers that are not conjugates?
3. What is the significance of the imaginary unit \(i\) in mathematics?
4. Can you apply this formula for any conjugate pair of complex numbers?
5. How do you represent complex numbers on the complex plane?

**Tip:** When multiplying complex conjugates, the imaginary parts cancel out, leaving a real number as the result.

Learn how to multiply complex conjugates using the formula (a + bi)(a - bi) = a^2 - (bi)^2. This detailed solution explains the multiplication of (5 + 2i)(5 - 2i), resulting in a real number 29. Suitable for high school students.

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