(3-4i)(6i+7)-(2-3i)

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Expressions: (3-4i)(6i+7)-(2-3i)

This article will guide you through simplifying the complex expression (3-4i)(6i+7)-(2-3i).

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 product:

   • We begin by expanding the first part of the expression, (3-4i)(6i+7), using the distributive property (also known as FOIL): (3-4i)(6i+7) = (3 * 6i) + (3 * 7) + (-4i * 6i) + (-4i * 7) = 18i + 21 - 24i² - 28i

2. Simplify using i² = -1:

   • Replace i² with -1: = 18i + 21 - 24(-1) - 28i = 18i + 21 + 24 - 28i

3. Combine real and imaginary terms:

   • Group the real terms and the imaginary terms: = (21 + 24) + (18 - 28)i = 45 - 10i

4. Subtract the second complex number:

   • Now, subtract the second complex number (2-3i) from the simplified result: = (45 - 10i) - (2 - 3i) = 45 - 10i - 2 + 3i

5. Combine real and imaginary terms again:

   • Group the real terms and the imaginary terms: = (45 - 2) + (-10 + 3)i

6. Final simplification:

   • Simplify the expression: = 43 - 7i

Conclusion

Therefore, the simplified form of the complex expression (3-4i)(6i+7)-(2-3i) is 43 - 7i.