Finding A^2 + B^2 + C^2 Given Ab = 6, Bc = 6, And Ca = 4
Introduction
In this article, we delve into an intriguing mathematical problem that involves finding the value of the expression a² + b² + c² given the relationships ab = 6, bc = 6, and ca = 4. This problem, while seemingly straightforward, requires a strategic approach and a solid understanding of algebraic manipulations. We will explore the steps involved in solving this problem, providing a comprehensive explanation that will benefit both students and math enthusiasts alike. The problem serves as a good exercise in applying fundamental algebraic principles and highlights the interconnectedness of mathematical concepts. Let's embark on this mathematical journey and unravel the solution together. To fully appreciate the solution, it's helpful to have a strong grasp of algebraic identities and techniques, such as substitution and equation manipulation. This problem not only tests your algebraic skills but also your ability to think critically and apply mathematical concepts in a creative way. We will break down the problem into manageable steps, ensuring that each stage is clearly explained. Understanding the logic behind each step is crucial for solving similar problems in the future. Moreover, we will emphasize the importance of checking your work to avoid common errors. This problem is a testament to the beauty of mathematics and how seemingly complex problems can be solved with the right approach and understanding.
Problem Statement
Our primary goal is to determine the value of the expression a² + b² + c²*, where a, b, and c are real numbers, given the following system of equations:
- ab = 6
- bc = 6
- ca = 4
This problem is a classic example of how algebraic relationships can be used to find specific values. It's important to note that we are not necessarily trying to find the individual values of a, b, and c. Instead, we are focused on finding the value of a particular expression involving these variables. This requires a slightly different approach than solving for individual variables. The given equations provide us with crucial information about the products of these variables, which we can then use to manipulate and find the desired sum of squares. This problem showcases the power of algebraic manipulation and how it can simplify complex problems into manageable steps. Before we proceed with the solution, it's essential to understand the relationships between the variables and the target expression. This understanding will guide our approach and help us make informed decisions throughout the solution process. The key to solving this problem lies in recognizing the connections between the given equations and how they relate to the expression a² + b² + c². By carefully analyzing these relationships, we can develop a strategy that will lead us to the correct answer.
Solution
Step 1: Multiplying the Equations
To begin, we multiply the three given equations together:
(ab) (bc) (ca) = 6 * 6 * 4
This simplifies to:
a² b² c² = 144
Taking the square root of both sides, we get:
abc = ±12
This step is crucial because it establishes a relationship between all three variables. By multiplying the equations, we've created a new equation that involves the product of a, b, and c. This product, abc, will be instrumental in finding the individual squares of the variables. The plus-minus sign (±) indicates that there are two possible values for abc: 12 and -12. This is an important consideration because it will affect the subsequent steps. The act of taking the square root introduces both positive and negative solutions, which we must account for. Ignoring the negative solution could lead to an incomplete or incorrect answer. This step highlights the importance of paying attention to mathematical details and considering all possibilities. The value of abc will serve as a bridge between the given equations and the expression we want to find, a² + b² + c². By connecting these pieces of information, we can move closer to the solution.
Step 2: Finding Individual Squares
Now, we can use the value of abc to find a², b², and c² individually. We will consider both cases, abc = 12 and abc = -12.
Case 1: abc = 12
- a² = (abc) / (bc) = 12 / 6 = 2
- b² = (abc) / (ca) = 12 / 4 = 3
- c² = (abc) / (ab) = 12 / 6 = 2
Case 2: abc = -12
- a² = (abc) / (bc) = -12 / 6 = -2
- b² = (abc) / (ca) = -12 / 4 = -3
- c² = (abc) / (ab) = -12 / 6 = -2
In this step, we leverage the value of abc that we found in the previous step. By dividing abc by each of the given equations (bc, ca, and ab), we can isolate the square of each variable. This is a clever application of algebraic manipulation and demonstrates how interconnected the equations are. The two cases arise from the two possible values of abc: 12 and -12. It's crucial to consider both cases because they lead to different values for the squares of the variables. In the first case, where abc = 12, we obtain positive values for a², b², and c². However, in the second case, where abc = -12, we encounter negative values. This is an important observation because it suggests that the original variables, a, b, and c, might be complex numbers. For the scope of this problem, we need to carefully evaluate what these negative values imply for the final solution. The division operation in this step is a key algebraic technique that allows us to isolate the squares of the variables. It's a direct application of the properties of equations and how they can be manipulated to reveal desired information. This step is a critical link in the solution process, connecting the product abc to the individual squares of the variables.
Step 3: Calculating a² + b² + c²
Now, we can calculate a² + b² + c² for both cases.
Case 1: abc = 12
a² + b² + c² = 2 + 3 + 2 = 7
Case 2: abc = -12
a² + b² + c² = -2 + (-3) + (-2) = -7
This is the final step in solving the problem. We simply add the values of a², b², and c² that we found in the previous step for each case. This step is straightforward but requires careful attention to the signs of the numbers. In the first case, where abc = 12, we obtain a positive value of 7 for a² + b² + c². However, in the second case, where abc = -12, we obtain a negative value of -7. This result is interesting because it shows that the value of the expression a² + b² + c² depends on the sign of abc. It's essential to note that the negative value in the second case arises from the negative values of a², b², and c². This implies that the variables a, b, and c could be imaginary numbers in this case. The addition operation in this step is a fundamental arithmetic operation, but it's crucial to perform it accurately to arrive at the correct answer. This step brings together all the previous steps and provides the final solution to the problem. The two possible values for a² + b² + c² highlight the importance of considering all possible cases and the nuances of algebraic solutions.
Final Answer
Therefore, the possible values for a² + b² + c² are 7 and -7.
This is the final conclusion of our solution. We have successfully found the possible values of a² + b² + c² given the initial conditions. The fact that we have two possible answers underscores the importance of considering all possible scenarios in mathematical problem-solving. The solution process involved several key algebraic techniques, including multiplying equations, taking square roots, and substituting values. Each step was carefully explained to provide a clear and comprehensive understanding of the solution. This problem serves as an excellent example of how algebraic principles can be applied to solve complex problems. It also highlights the importance of paying attention to details and considering all possible cases. The final answer, 7 and -7, represents the complete solution to the problem and showcases the power of algebraic manipulation and logical reasoning.
Conclusion
In this article, we successfully determined the possible values of the expression a² + b² + c² given the conditions ab = 6, bc = 6, and ca = 4. We navigated through the problem by strategically multiplying the equations, finding the possible values of abc, and subsequently calculating the individual squares of a, b, and c. The final step involved summing these squares, leading us to the solutions 7 and -7. This problem underscores the elegance of algebraic problem-solving and highlights the importance of considering multiple cases. The solution process involved a series of logical steps, each building upon the previous one. This systematic approach is crucial for tackling complex mathematical problems. The fact that we obtained two possible solutions emphasizes the importance of thoroughness and attention to detail in mathematical reasoning. This problem not only tests algebraic skills but also promotes critical thinking and problem-solving abilities. By breaking down the problem into manageable steps, we were able to arrive at the correct solution. This approach can be applied to a wide range of mathematical problems. The problem serves as a valuable learning experience and demonstrates the interconnectedness of mathematical concepts. We hope this detailed explanation has provided a clear understanding of the solution process and has inspired you to explore more mathematical challenges.
