%5E2.jpeg "(x-2i)^2")
Expanding (x - 2i)²
In mathematics, expanding expressions like (x - 2i)² often involves understanding complex numbers and their operations. Here's how we can approach this expansion:
Understanding Complex Numbers
Complex numbers involve the imaginary unit, i, where i² = -1. They are expressed in the form a + bi, where a and b are real numbers.
Expanding the Expression
To expand (x - 2i)², we can use the FOIL method (First, Outer, Inner, Last) or simply apply the square of a binomial pattern:
(a - b)² = a² - 2ab + b²
Applying the pattern:
- Square the first term: x²
- Multiply the two terms and double the product: -2 * x * 2i = -4ix
- Square the second term: (2i)² = 4i² = 4(-1) = -4
Combining the terms:
(x - 2i)² = x² - 4ix - 4
Therefore, the expanded form of (x - 2i)² is x² - 4ix - 4.
Simplifying the Expression
While the expression is expanded, it can be further simplified by rearranging the terms:
x² - 4ix - 4
This form is commonly used to express the result in standard complex number form (a + bi), where the real part is x² - 4 and the imaginary part is -4x.
Conclusion
Expanding (x - 2i)² involves understanding the properties of complex numbers and applying the appropriate expansion method. By using the FOIL method or the square of a binomial pattern, we can arrive at the expanded form of the expression, which can then be simplified to a standard complex number form.