## Multiplying Complex Numbers: (5 + 2i)(5 - 2i)

This article explores the multiplication of complex numbers, specifically focusing on the product of (5 + 2i) and (5 - 2i).

### Understanding Complex Numbers

Complex numbers are numbers that can be expressed in the form **a + bi**, where **a** and **b** are real numbers and **i** is the imaginary unit, defined as the square root of -1.

### Multiplying Complex Numbers

To multiply complex numbers, we use the distributive property, similar to multiplying binomials:

(a + bi)(c + di) = ac + adi + bci + bdi²

Since i² = -1, we can simplify this to:

(a + bi)(c + di) = (ac - bd) + (ad + bc)i

### Applying the Formula

Let's apply this formula to our problem:

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

Simplifying the expression:

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

Therefore, the product of (5 + 2i) and (5 - 2i) is:

**(5 + 2i)(5 - 2i) = 29**

### Key Observation:

Notice that the product of (5 + 2i) and (5 - 2i) is a real number. This is because (5 - 2i) is the **complex conjugate** of (5 + 2i). The product of a complex number and its conjugate always results in a real number.

### Conclusion

Multiplying complex numbers requires careful application of the distributive property and recognizing the special case of multiplying a number by its conjugate. In the case of (5 + 2i)(5 - 2i), the product simplifies to a real number, highlighting the relationship between complex numbers and their conjugates.