2.jpeg "(3−2i)2")
Simplifying (3 - 2i)<sup>2</sup>
This article will explore the simplification of the expression (3 - 2i)<sup>2</sup>, focusing on the fundamentals of complex number arithmetic.
Understanding Complex Numbers
A complex number is a number 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 (i<sup>2</sup> = -1).
Expanding the Expression
To simplify (3 - 2i)<sup>2</sup>, we can expand it using the distributive property (FOIL method):
(3 - 2i)<sup>2</sup> = (3 - 2i)(3 - 2i)
Expanding the product, we get:
(3 - 2i)(3 - 2i) = 3(3) + 3(-2i) - 2i(3) - 2i(-2i)
Simplifying further:
= 9 - 6i - 6i + 4i<sup>2</sup>
Replacing i<sup>2</sup> with -1
Recall that i<sup>2</sup> = -1. Substituting this value into the expression:
= 9 - 6i - 6i + 4(-1)
Combining Real and Imaginary Terms
Combining like terms:
= 9 - 4 - 6i - 6i
= 5 - 12i
Final Result
Therefore, the simplified form of (3 - 2i)<sup>2</sup> is 5 - 12i.