(5%2B2i)-in-a%2Bbi-form.jpeg "(5-2i)(5+2i) In A+bi Form")
Simplifying Complex Numbers: (5-2i)(5+2i)
This article aims to demonstrate how to simplify the product of two complex numbers, (5-2i) and (5+2i), and express the result in the standard form of a complex number (a+bi).
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
- i is the imaginary unit, defined as the square root of -1 (i² = -1).
Simplifying the Product
We can simplify the product (5-2i)(5+2i) using the distributive property (or FOIL method):
(5-2i)(5+2i) = 5(5+2i) - 2i(5+2i)
Expanding the expression, we get:
= 25 + 10i - 10i - 4i²
Since i² = -1, we can substitute:
= 25 + 10i - 10i - 4(-1)
Combining like terms:
= 25 + 4
= 29
Therefore, the product (5-2i)(5+2i) simplifies to 29.
Result in a+bi form
The result, 29, can be expressed in the standard form of a complex number as 29 + 0i.
Key Points
- The product of a complex number and its conjugate always results in a real number.
- The conjugate of a complex number is formed by changing the sign of the imaginary part.
- Complex numbers are often used in various fields like physics, engineering, and mathematics.
This example demonstrates the process of simplifying complex numbers and expressing them in the standard form (a+bi). Understanding complex numbers and their operations is crucial for solving various mathematical problems.