(4−3xi)(5+3i)+(4+3xi)(5+3i)

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Expressions: (4−3xi)(5+3i)+(4+3xi)(5+3i)

This article will guide you through the process of simplifying the complex expression: (4−3xi)(5+3i)+(4+3xi)(5+3i).

Understanding Complex Numbers

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.

Simplifying the Expression

1. Distribute: We start by expanding both expressions using the distributive property (also known as FOIL):

   (4−3xi)(5+3i) = (4 * 5) + (4 * 3i) + (-3xi * 5) + (-3xi * 3i) (4+3xi)(5+3i) = (4 * 5) + (4 * 3i) + (3xi * 5) + (3xi * 3i)

2. Simplify: Next, we simplify each term:

   (4−3xi)(5+3i) = 20 + 12i - 15xi - 9i² (4+3xi)(5+3i) = 20 + 12i + 15xi - 9i²

3. Substitute i² = -1: Remember that i² is equal to -1. Substitute this into the expressions:

   (4−3xi)(5+3i) = 20 + 12i - 15xi + 9 (4+3xi)(5+3i) = 20 + 12i + 15xi + 9

4. Combine like terms: Group the real and imaginary terms together:

   (4−3xi)(5+3i) = (20 + 9) + (12 - 15x)i (4+3xi)(5+3i) = (20 + 9) + (12 + 15x)i

5. Add the expressions: Now, we can add the simplified expressions:

   (20 + 9) + (12 - 15x)i + (20 + 9) + (12 + 15x)i = 49 + 24i

Final Result

Therefore, the simplified form of the expression (4−3xi)(5+3i)+(4+3xi)(5+3i) is 49 + 24i.