Solving √(-3r + 7) = 7i A Step-by-Step Solution
This article delves into the process of solving the equation √(-3r + 7) = 7i, where 'r' is a variable and 'i' represents the imaginary unit (√-1). This type of equation involves complex numbers and radicals, requiring a careful approach to arrive at the correct solution. We will break down the steps involved, explain the underlying concepts, and provide a clear, comprehensive guide for anyone looking to understand and solve this kind of problem. Understanding how to manipulate equations involving square roots and imaginary numbers is crucial in various fields of mathematics, physics, and engineering. Therefore, mastering this process not only solves the specific equation but also enhances your broader mathematical skills.
Understanding the Fundamentals
Before we jump into the solution, let's recap some essential concepts. The equation √(-3r + 7) = 7i combines two key elements: square roots and imaginary numbers. A square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3 because 3 * 3 = 9. However, when dealing with negative numbers under a square root, we enter the realm of imaginary numbers. The imaginary unit, denoted as 'i', is defined as the square root of -1 (i = √-1). This concept is fundamental to complex numbers, which have the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit. Complex numbers extend the real number system and are vital in many areas of mathematics and its applications.
In this context, the left side of our equation √(-3r + 7) involves a square root, and the right side, 7i, is an imaginary number. Our goal is to find the value of 'r' that satisfies this equation. This involves isolating 'r' by performing valid algebraic operations on both sides of the equation. The process will include squaring both sides to eliminate the square root and then solving for 'r'. It's important to remember that squaring both sides can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original one. Therefore, we must verify our solution by plugging it back into the original equation.
Step-by-Step Solution
Now, let’s dive into the step-by-step solution of the equation √(-3r + 7) = 7i. This will involve several algebraic manipulations, keeping in mind the properties of square roots and imaginary numbers. We'll meticulously detail each step to ensure clarity.
Step 1: Squaring Both Sides
The first step to eliminate the square root is to square both sides of the equation. This operation is based on the principle that if two expressions are equal, their squares are also equal. So, if √(-3r + 7) = 7i, then squaring both sides gives us:
(√(-3r + 7))² = (7i)²
Squaring the left side cancels out the square root, leaving us with:
-3r + 7
On the right side, we have (7i)². Recall that i² = -1. Thus,
(7i)² = 7² * i² = 49 * (-1) = -49
Our equation now becomes:
-3r + 7 = -49
This step is crucial because it transforms the original equation with a square root into a simpler linear equation involving 'r'.
Step 2: Isolating the Term with 'r'
The next step is to isolate the term containing 'r' on one side of the equation. We can do this by subtracting 7 from both sides:
-3r + 7 - 7 = -49 - 7
This simplifies to:
-3r = -56
By subtracting 7 from both sides, we have successfully isolated the term with 'r' on the left side. This brings us closer to finding the value of 'r'.
Step 3: Solving for 'r'
To solve for 'r', we need to divide both sides of the equation by -3:
-3r / -3 = -56 / -3
This gives us:
r = 56/3
So, we have found a potential solution for 'r'. However, it's essential to verify this solution by substituting it back into the original equation, especially since we squared both sides in the first step, which could introduce extraneous solutions.
Step 4: Verification
To verify our solution, we substitute r = 56/3 back into the original equation √(-3r + 7) = 7i:
√(-3(56/3) + 7) = 7i
First, simplify the expression inside the square root:
-3(56/3) = -56
So, we have:
√(-56 + 7) = 7i
√(-49) = 7i
Since √-49 = √(49 * -1) = √(49) * √(-1) = 7i, the equation holds true:
7i = 7i
This verification step confirms that r = 56/3 is indeed a valid solution to the original equation.
Final Answer
Therefore, the solution to the equation √(-3r + 7) = 7i is:
r = 56/3
This solution satisfies the original equation, making it the correct answer.
Common Mistakes and How to Avoid Them
When solving equations involving square roots and imaginary numbers, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and arrive at the correct solution more efficiently. Let's discuss some of these mistakes and how to prevent them.
Mistake 1: Forgetting to Square Both Sides Completely
One common mistake is not squaring the entire term on each side of the equation. For example, when squaring (7i), some might forget to square the 7, resulting in an incorrect simplification. Remember that (7i)² = 7² * i² = 49 * -1 = -49. Always ensure that every part of the term is properly squared.
Mistake 2: Incorrectly Handling Imaginary Unit i
The imaginary unit 'i' has a unique property: i² = -1. Misunderstanding or forgetting this property can lead to significant errors. For instance, when squaring 7i, one must remember that i² becomes -1, changing the sign of the term. Always replace i² with -1 whenever it appears in your calculations.
Mistake 3: Not Verifying the Solution
Squaring both sides of an equation can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original one. Therefore, it is crucial to verify your solution by substituting it back into the original equation. If the solution does not satisfy the original equation, it is an extraneous solution and should be discarded.
Mistake 4: Errors in Algebraic Manipulation
Algebraic manipulation errors, such as incorrect addition, subtraction, multiplication, or division, can lead to incorrect solutions. To avoid these, take your time and double-check each step. Pay close attention to signs (positive and negative) and ensure you are performing the same operation on both sides of the equation to maintain balance.
Mistake 5: Misunderstanding Square Roots of Negative Numbers
The square root of a negative number is an imaginary number. For instance, √-49 is not a real number; it is 7i. Misinterpreting this can lead to errors in equations involving square roots and imaginary numbers. Always remember that √-a = i√a, where a is a positive real number.
How to Avoid These Mistakes
- Double-Check Your Work: Always review each step of your calculation to catch any errors in algebraic manipulation.
- Remember the Properties of i: Keep in mind that i² = -1 and use this property correctly in your calculations.
- Verify Your Solution: Always substitute your solution back into the original equation to ensure it is valid.
- Practice Regularly: The more you practice solving equations involving square roots and imaginary numbers, the more comfortable and proficient you will become.
- Write Neatly and Organize Your Work: A clear and organized approach can help you avoid errors and make it easier to review your steps.
By being aware of these common mistakes and following these tips, you can significantly improve your accuracy in solving equations involving square roots and imaginary numbers.
Applications of Complex Numbers
Complex numbers, which include imaginary numbers, are not just abstract mathematical concepts; they have a wide range of practical applications in various fields. Understanding these applications can help appreciate the importance of mastering complex number manipulations, such as solving equations like the one we addressed. Let's explore some key areas where complex numbers play a crucial role.
Electrical Engineering
In electrical engineering, complex numbers are fundamental for analyzing alternating current (AC) circuits. AC circuits involve sinusoidal voltages and currents that can be represented using complex numbers. Impedance, which is the opposition to the flow of current in an AC circuit, is a complex quantity that includes both resistance and reactance (due to inductors and capacitors). By using complex numbers, engineers can simplify the analysis of AC circuits, calculate currents, voltages, and power, and design efficient electrical systems. The use of complex numbers allows for a more straightforward representation and manipulation of these quantities compared to using trigonometric functions.
Quantum Mechanics
Quantum mechanics, the theory that governs the behavior of matter at the atomic and subatomic levels, heavily relies on complex numbers. The wave function, which describes the state of a quantum system, is a complex-valued function. The square of the magnitude of the wave function gives the probability density of finding a particle at a particular location. Complex numbers are essential for representing the wave-like nature of particles and for describing phenomena such as quantum entanglement and superposition. Without complex numbers, many of the fundamental principles of quantum mechanics would be impossible to formulate.
Signal Processing
Signal processing, which involves the analysis and manipulation of signals such as audio, images, and video, makes extensive use of complex numbers. The Fourier transform, a crucial tool in signal processing, transforms a signal from the time domain to the frequency domain, and it is defined using complex exponentials. By representing signals in the frequency domain, signal processing algorithms can filter, compress, and analyze signals more effectively. Complex numbers also play a role in digital signal processing (DSP) for applications such as audio compression, image enhancement, and telecommunications.
Control Systems
Control systems, which are used to regulate the behavior of dynamic systems, often employ complex numbers in their analysis and design. The stability and performance of a control system can be analyzed using techniques such as the Laplace transform, which involves complex variables. Control engineers use complex numbers to design feedback controllers that ensure systems operate stably and meet performance requirements. Applications of control systems range from industrial automation to aerospace engineering.
Fluid Dynamics
In fluid dynamics, complex numbers are used to represent two-dimensional fluid flows. The velocity potential and stream function, which describe the flow of a fluid, can be combined into a complex potential. This complex representation simplifies the analysis of fluid flows, particularly in aerodynamics and hydrodynamics. Complex analysis techniques can be used to solve problems related to lift, drag, and flow patterns around objects.
Fractals
Fractals, which are geometric shapes with self-similar patterns, are often defined using complex numbers. The Mandelbrot set, a famous example of a fractal, is defined based on the behavior of complex numbers under repeated iterations of a simple equation. Complex numbers provide a natural framework for generating and studying fractals, which have applications in computer graphics, image compression, and modeling natural phenomena.
Other Applications
Besides the above-mentioned applications, complex numbers also find use in:
- Geophysics: Analyzing seismic waves.
- Medical Imaging: Processing MRI and CT scan data.
- Cryptography: Developing encryption algorithms.
- Mathematics: Various areas, including complex analysis, number theory, and algebra.
These diverse applications highlight the pervasive nature of complex numbers in science and engineering. The ability to work with complex numbers is a valuable skill for anyone pursuing a career in these fields. Mastering the fundamentals of complex number manipulations, such as solving equations, is an essential step towards understanding and applying these concepts in real-world problems.
Conclusion
In conclusion, solving the equation √(-3r + 7) = 7i is a process that requires a solid understanding of square roots, imaginary numbers, and algebraic manipulations. We have demonstrated a step-by-step approach, starting with squaring both sides to eliminate the square root, isolating the term with 'r', solving for 'r', and finally, verifying the solution to avoid extraneous roots. The correct solution we found is r = 56/3. We also discussed common mistakes to avoid, such as forgetting to square the entire term, mishandling the imaginary unit 'i', not verifying the solution, and making algebraic errors. Being mindful of these pitfalls can help ensure accuracy in your calculations.
Furthermore, we explored the wide range of applications of complex numbers in various fields, including electrical engineering, quantum mechanics, signal processing, control systems, fluid dynamics, and fractal geometry. These applications underscore the practical importance of complex numbers and the value of mastering their manipulation.
By understanding the methods for solving equations involving complex numbers and appreciating their applications, you can enhance your mathematical skills and prepare yourself for more advanced topics in science and engineering. Practice is key to mastering these concepts, so continue to work through similar problems to build your proficiency and confidence. Whether you are a student learning the basics or a professional applying these concepts in your work, a strong foundation in complex numbers will undoubtedly prove invaluable.
