Solving For A And B In Complex Number Equations

Introduction

In the realm of mathematics, complex numbers often present intriguing challenges. This article delves into the process of solving an equation involving complex numbers to determine the values of the real and imaginary parts. We will explore how to simplify expressions with square roots of negative numbers, perform algebraic manipulations, and ultimately arrive at the solution in the form of a + bi, where a and b are real numbers. By understanding the fundamental concepts and techniques, you can confidently tackle similar problems and expand your knowledge of complex number arithmetic.

In this article, we will thoroughly analyze the equation (4+√-49)-2(√((-4)²)+√-324)=a+bi, and step-by-step find the accurate values for a and b. We will discuss the procedures to be followed when simplifying complex numbers, and will discuss in detail each mathematical operation, so that the reader can clearly understand. This approach will enhance not only comprehension but also problem-solving skills for similar mathematical challenges. By the end of this explanation, readers will achieve the ability to methodically demystify and solve equations involving complex numbers, enriching their mathematical knowledge.

Breaking Down the Equation: Step-by-Step Solution

To solve the equation (4+√-49)-2(√((-4)²)+√-324)=a+bi, we will proceed step by step, simplifying each part of the equation until we isolate the real and imaginary components. This meticulous approach will enable us to accurately determine the values of a and b. In this section, we will thoroughly discuss each step, clarifying the underlying principles and strategies used to manipulate complex numbers effectively. By dividing the process into manageable parts, we ensure clarity and facilitate a deeper understanding of the solution.

1. Simplify the Square Roots of Negative Numbers

Our initial focus is on simplifying the square roots of negative numbers. Remember, the square root of -1 is defined as i, the imaginary unit. We will apply this principle to √-49 and √-324. Breaking down these roots will allow us to express them in terms of i, thereby simplifying the equation and making it easier to solve. The manipulation of square roots of negative numbers is crucial in complex number arithmetic, and mastering this skill will be invaluable for solving more complex equations.

• √-49: We can rewrite √-49 as √(49 * -1), which equals √(49) * √(-1). Since √(49) is 7 and √(-1) is i, the simplified form is 7i. This transformation is crucial because it allows us to transition from a square root of a negative number to an imaginary number, making it more compatible with standard complex number operations.
• √-324: Similarly, we express √-324 as √(324 * -1), which is √(324) * √(-1). As √(324) is 18 and √(-1) is i, this simplifies to 18i. This simplification is vital for handling the imaginary part of complex numbers and allows us to perform further calculations with precision.

2. Simplify the Square Root of (-4)²

Next, we simplify the term √((-4)²). When squaring a negative number, the result is positive. Therefore, (-4)² equals 16. The square root of 16 is 4. This step is important as it helps in converting the expression to a real number, further clarifying the equation. Simplifying squares under the square root is a common technique in complex number calculations, and mastering this step ensures accuracy in your solutions.

• √((-4)²): Here, (-4)² equals 16. Thus, √((-4)²) becomes √16, which simplifies to 4. This conversion from the square of a negative number to a positive number and its square root is a basic yet crucial step in simplifying complex mathematical expressions.

3. Substitute the Simplified Terms Back into the Equation

Now, we substitute the simplified terms back into the original equation. Replacing √-49 with 7i, √((-4)²) with 4, and √-324 with 18i gives us a clearer equation to work with. This substitution is a critical step, as it brings us closer to isolating the real and imaginary parts of the equation. By plugging in the simplified terms, we transition from an equation with complex roots to one that is easier to manage and solve.

• Original Equation: (4+√-49)-2(√((-4)²)+√-324)=a+bi
• Substituting: (4 + 7i) - 2(4 + 18i) = a + bi

4. Distribute and Combine Like Terms

The next step is to distribute the -2 across the terms inside the parentheses and then combine like terms. This algebraic manipulation will help us separate the real and imaginary components of the equation. Distributing and combining like terms is a fundamental technique in algebra and is crucial for simplifying complex expressions. This step prepares the equation for the final isolation of real and imaginary parts.

• Distribute -2: (4 + 7i) - 2(4) - 2(18i) = a + bi simplifies to 4 + 7i - 8 - 36i = a + bi.
• Combine like terms: (4 - 8) + (7i - 36i) = a + bi. This further simplifies to -4 - 29i = a + bi. Combining like terms is crucial for grouping real and imaginary parts, paving the way for the final determination of a and b.

5. Equate Real and Imaginary Parts

Finally, we equate the real and imaginary parts on both sides of the equation. The real part on the left side is -4, and the imaginary part is -29. This allows us to directly determine the values of a and b. Equating real and imaginary parts is a fundamental concept in complex number algebra, and it is the key to solving for unknown variables in complex equations. This step provides a straightforward method to find the solution by matching coefficients.

• Equating real parts: a = -4
• Equating imaginary parts: b = -29

Final Answer: The Values of a and b

After simplifying the equation (4+√-49)-2(√((-4)²)+√-324)=a+bi step by step, we have successfully determined the values of a and b. The meticulous approach of simplifying square roots, substituting terms, distributing, combining like terms, and equating real and imaginary parts has led us to the final answer. This demonstrates a robust methodology for handling complex number equations.

• The value of a is -4. This represents the real component of the complex number resulting from the equation.
• The value of b is -29. This signifies the imaginary component of the complex number.

Therefore, the solution to the equation is a = -4 and b = -29. By dissecting the equation and addressing each component individually, we have provided a comprehensive solution. This process not only solves the problem but also enhances understanding of complex number operations. This thorough breakdown ensures that readers can follow along and apply similar methods to future problems, solidifying their grasp of the subject matter.

Conclusion

In summary, solving equations with complex numbers requires a systematic approach, including simplifying square roots of negative numbers, substituting terms, distributing, combining like terms, and equating real and imaginary parts. By following these steps, we accurately determined that a is -4 and b is -29 in the equation (4+√-49)-2(√((-4)²)+√-324)=a+bi. Understanding these methods enhances your ability to tackle complex number problems effectively. The ability to methodically solve such equations is crucial for various fields, including engineering, physics, and advanced mathematics. This in-depth exploration not only provides the solution but also equips the reader with the tools necessary to excel in similar mathematical scenarios. Continued practice and application of these techniques will build confidence and proficiency in complex number arithmetic.

This journey through complex numbers underscores the significance of methodical problem-solving in mathematics. Each step, from simplifying square roots to equating real and imaginary parts, builds upon the previous one, leading to a clear and accurate solution. The mastery of these techniques not only empowers you to solve equations but also deepens your understanding of the broader mathematical landscape. Embrace the challenges that complex numbers present, and you'll find that they unlock new dimensions in your mathematical thinking and problem-solving capabilities.