(3 – 2i)(5 + 4i) – (3 – 4i)^2

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Expressions

This article will walk you through the process of simplifying the complex expression: (3 – 2i)(5 + 4i) – (3 – 4i)^2.

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.

Step-by-Step Simplification

Let's break down the expression step-by-step:

1. Expanding the first product: (3 – 2i)(5 + 4i) = (3 * 5) + (3 * 4i) + (-2i * 5) + (-2i * 4i) = 15 + 12i - 10i - 8i^2

2. Expanding the second product: (3 - 4i)^2 = (3 - 4i)(3 - 4i) = (3 * 3) + (3 * -4i) + (-4i * 3) + (-4i * -4i) = 9 - 12i - 12i + 16i^2

3. Substituting i^2 with -1: Remember that i^2 = -1. Therefore, we can substitute:

   • 15 + 12i - 10i - 8i^2 = 15 + 12i - 10i + 8
   • 9 - 12i - 12i + 16i^2 = 9 - 12i - 12i - 16

4. Combining like terms:

   • (15 + 8) + (12i - 10i) = 23 + 2i
   • (9 - 16) + (-12i - 12i) = -7 - 24i

5. Final Result: Finally, we subtract the two simplified expressions: (23 + 2i) - (-7 - 24i) = 23 + 2i + 7 + 24i = 30 + 26i

Conclusion

Therefore, the simplified form of the complex expression (3 – 2i)(5 + 4i) – (3 – 4i)^2 is 30 + 26i.