(1 + 2i)(2 + I) + = 5(2 + I)

Jasonbradley

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Jun 16, 2024

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Solving the Complex Equation: (1 + 2i)(2 + i) + = 5(2 + i)

This article will walk through the steps of solving the complex equation (1 + 2i)(2 + i) + = 5(2 + i). We will utilize the principles of complex number arithmetic to simplify and isolate the unknown variable.

Understanding the Basics of 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).

Key Operations with Complex Numbers:

• Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
• Subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i
• Multiplication: (a + bi)(c + di) = (ac - bd) + (ad + bc)i

Solving the Equation

1. Expand the product:

   Start by expanding the left side of the equation using the distributive property of multiplication:

   (1 + 2i)(2 + i) = (1 * 2 + 1 * i + 2i * 2 + 2i * i) = (2 + i + 4i - 2) = 5i

   Now the equation becomes: 5i + = 5(2 + i)

2. Simplify the right side:

   Distribute the 5 on the right side:

   5(2 + i) = 10 + 5i

   The equation now becomes: 5i + = 10 + 5i

3. Isolate the unknown variable:

   To isolate the unknown variable, subtract 5i from both sides:

   5i + - 5i = 10 + 5i - 5i

   This leaves us with: = 10

4. Solution:

   The solution to the equation is ** = 10**.

Conclusion

We have successfully solved the complex equation (1 + 2i)(2 + i) + = 5(2 + i) by applying the principles of complex number arithmetic and simplifying the equation step-by-step. The solution is = 10.