(7-3i)+(x-2i)^2-(4i+2x^2) In A+bi Form

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Expressions: (7-3i) + (x-2i)² - (4i + 2x²)

This article will guide you through simplifying the complex expression (7-3i) + (x-2i)² - (4i + 2x²) into the standard form a + bi.

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 (i² = -1).

Step-by-Step Simplification

1. Expand the Square: Begin by expanding the squared term (x - 2i)²: (x - 2i)² = x² - 4xi + 4i²

2. Substitute i²: Remember that i² = -1. Substitute this into the expanded term: x² - 4xi + 4i² = x² - 4xi - 4

3. Combine Terms: Combine all the terms: (7 - 3i) + (x² - 4xi - 4) - (4i + 2x²) = 7 - 3i + x² - 4xi - 4 - 4i - 2x²

4. Rearrange and Group: Rearrange the terms to group the real and imaginary parts: (7 - 4 - 2x²) + (-3i - 4i - 4xi) = (3 - 2x²) + (-7 - 4x)i

Final Result

Therefore, the simplified form of the complex expression (7-3i) + (x-2i)² - (4i + 2x²) is (3 - 2x²) + (-7 - 4x)i.

This is now in the standard form a + bi, where:

• a = 3 - 2x²
• b = -7 - 4x