Simplifying Complex Expressions: (73i) + (x2i)²  (4i + 2x²)
This article will guide you through simplifying the complex expression (73i) + (x2i)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).
StepbyStep Simplification

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

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

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

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 (73i) + (x2i)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