(-7+3i)^2+(-2+8i)

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Expressions

This article will guide you through simplifying the expression (-7 + 3i)^2 + (-2 + 8i).

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 (i.e., i² = -1).

Simplifying the Expression

1. Expand the square:

   • (-7 + 3i)² = (-7 + 3i)(-7 + 3i)
   • Using the FOIL method (First, Outer, Inner, Last):
     • (-7)(-7) + (-7)(3i) + (3i)(-7) + (3i)(3i)
     • 49 - 21i - 21i + 9i²
     • 49 - 42i - 9 (since i² = -1)
     • 40 - 42i

2. Combine the terms:

   • (40 - 42i) + (-2 + 8i)
   • (40 - 2) + (-42 + 8)i
   • 38 - 34i

The Final Result

Therefore, the simplified form of the expression (-7 + 3i)² + (-2 + 8i) is 38 - 34i.