## Squaring Complex Numbers: Exploring (5 + 2i)^2

This article delves into the process of squaring the complex number (5 + 2i). We'll explore the concept of complex numbers, the rules for squaring them, and arrive at the final result.

### Understanding Complex Numbers

Complex numbers are numbers that extend the real number system by incorporating the imaginary unit, **i**, where **i² = -1**. They are expressed in the form **a + bi**, where **a** and **b** are real numbers.

### Squaring Complex Numbers

Squaring a complex number involves multiplying it by itself. The process is similar to squaring real numbers, but we need to keep in mind the properties of **i**.

### Calculating (5 + 2i)²

Let's break down the calculation step-by-step:

**(5 + 2i)² = (5 + 2i) * (5 + 2i)**

Expanding the product using the distributive property, we get:

**(5 + 2i)² = 5 * (5 + 2i) + 2i * (5 + 2i)**

Simplifying further:

**(5 + 2i)² = 25 + 10i + 10i + 4i²**

Since **i² = -1**, we can substitute:

**(5 + 2i)² = 25 + 10i + 10i + 4(-1)**

Combining real and imaginary terms:

**(5 + 2i)² = (25 - 4) + (10 + 10)i**

Therefore, the final result is:

**(5 + 2i)² = 21 + 20i**

### Conclusion

Squaring a complex number involves careful application of the distributive property and the fundamental property of the imaginary unit, **i² = -1**. By following these steps, we can efficiently calculate the square of any complex number. In this case, (5 + 2i)² simplifies to **21 + 20i**.