%5E2.jpeg "(3-6i)^2")
Squaring Complex Numbers: (3-6i)^2
This article explores the squaring of the complex number (3 - 6i). We'll use the fundamental properties of complex numbers to find the solution.
Understanding Complex Numbers
Complex numbers are numbers of the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit, defined as the square root of -1 (i² = -1).
Squaring (3 - 6i)
To square (3 - 6i), we multiply it by itself:
(3 - 6i)² = (3 - 6i) * (3 - 6i)
We can expand this using the distributive property (FOIL method):
(3 - 6i) * (3 - 6i) = 3 * 3 + 3 * (-6i) - 6i * 3 - 6i * (-6i)
Simplifying the terms:
= 9 - 18i - 18i + 36i²
Since i² = -1, we can substitute:
= 9 - 18i - 18i + 36(-1)
Combining the real and imaginary terms:
= 9 - 36 - 18i - 18i
= -27 - 36i
The Solution
Therefore, (3 - 6i)² = -27 - 36i.
This demonstrates that squaring a complex number results in another complex number, with both real and imaginary components.