%5E2.jpeg "(5-2i)^2")
Squaring Complex Numbers: (5-2i)^2
In mathematics, 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.
Squaring a complex number like (5-2i) involves multiplying it by itself. Let's break down the steps:
1. Expanding the Expression:
(5-2i)^2 = (5-2i)(5-2i)
2. Applying the FOIL Method:
FOIL stands for First, Outer, Inner, Last. It's a technique to multiply two binomials:
- First: 5 * 5 = 25
- Outer: 5 * -2i = -10i
- Inner: -2i * 5 = -10i
- Last: -2i * -2i = 4i^2
3. Simplifying using i^2 = -1:
Combining the terms and substituting i^2 with -1:
25 - 10i - 10i + 4(-1) = 25 - 20i - 4
4. Final Result:
Therefore, (5-2i)^2 simplifies to 21 - 20i.
In conclusion, squaring a complex number involves expanding the expression, applying the FOIL method, and simplifying using the property i^2 = -1. The result of (5-2i)^2 is 21 - 20i, another complex number in the form a + bi.