(3-4i)(1+5i)-(2-i)

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Expressions

This article will guide you through simplifying the complex expression: (3-4i)(1+5i) - (2-i)

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).

Simplifying the Expression

1. Expand the product: We begin by expanding the product (3-4i)(1+5i) using the distributive property (FOIL method):

   (3-4i)(1+5i) = 3(1) + 3(5i) - 4i(1) - 4i(5i) = 3 + 15i - 4i - 20i²

2. Substitute i² with -1: Since i² = -1, we can substitute it in the expression:

   3 + 15i - 4i - 20i² = 3 + 15i - 4i + 20

3. Combine real and imaginary terms: Now, combine the real terms (3 + 20) and the imaginary terms (15i - 4i):

   3 + 15i - 4i + 20 = 23 + 11i

4. Subtract (2-i): Finally, subtract (2-i) from the simplified expression:

   23 + 11i - (2-i) = 23 + 11i - 2 + i

5. Combine real and imaginary terms again: Combine the real terms (23 - 2) and the imaginary terms (11i + i):

   23 + 11i - 2 + i = 21 + 12i

Conclusion

Therefore, the simplified form of the complex expression (3-4i)(1+5i) - (2-i) is 21 + 12i.