In the captivating world of mathematics, we often encounter intriguing problems that require us to put on our thinking caps and embark on a journey of discovery. Today, we delve into one such problem that involves deciphering a numerical enigma. Our mission, should we choose to accept it, is to find the number(s) that satisfy a specific condition. Let us begin this mathematical expedition with zeal and determination, for the reward of intellectual satisfaction awaits us at the end.
Decoding the Problem Statement
The problem at hand presents us with a curious statement three times the square of the difference between a number and 4 is 192. This statement, while seemingly straightforward, holds the key to unlocking the numerical mystery we seek to solve. To effectively tackle this problem, we must first break down the statement into its constituent parts and translate them into mathematical expressions. This process of translation is akin to converting a sentence in a foreign language into our native tongue, allowing us to grasp the underlying meaning and context.
From Words to Equations
The first step in our translation process is to identify the unknown quantity, which in this case is the number we are trying to find. Let us denote this number by the variable x. Now, we can proceed to translate the verbal phrases into mathematical expressions. The phrase the difference between a number and 4 can be written as x - 4. Next, the square of the difference between a number and 4 becomes (x - 4)^2. Finally, three times the square of the difference between a number and 4 translates to 3(x - 4)^2. With this translation in hand, we can now express the entire problem statement as a mathematical equation:
3(x - 4)^2 = 192
This equation is the heart of our problem, the Rosetta Stone that will guide us to the solution. It encapsulates the essence of the problem statement in a concise and unambiguous form. Now that we have successfully translated the problem into an equation, we can proceed to the next phase of our mathematical journey: solving the equation.
Solving the Equation A Step-by-Step Guide
With our equation 3(x - 4)^2 = 192 firmly in hand, we are now ready to embark on the process of solving for the unknown number x. This process involves a series of algebraic manipulations designed to isolate x on one side of the equation. Each step we take is guided by the principles of mathematical equality, ensuring that we maintain the balance and integrity of the equation.
Step 1 Divide both sides by 3
Our first step is to simplify the equation by dividing both sides by 3. This operation removes the coefficient of 3 from the left side, making the equation more manageable. Performing this division, we obtain:
(x - 4)^2 = 64
Step 2 Take the square root of both sides
Next, we take the square root of both sides of the equation. This operation undoes the squaring on the left side, bringing us closer to isolating x. It is crucial to remember that when taking the square root, we must consider both positive and negative roots. This is because both the positive and negative square roots of a number, when squared, yield the original number. Applying the square root operation, we get:
x - 4 = ±8
Step 3 Isolate x
Now, we have two separate equations to solve: x - 4 = 8 and x - 4 = -8. To isolate x in each equation, we simply add 4 to both sides. This gives us:
x = 8 + 4 = 12
and
x = -8 + 4 = -4
Therefore, we have found two possible solutions for x: 12 and -4. These numbers satisfy the original equation, meaning they are the numbers we sought to find.
Verifying the Solutions The Moment of Truth
In the realm of mathematics, it is always prudent to verify our solutions. This practice serves as a safeguard against errors and ensures that our answers truly satisfy the conditions of the problem. To verify our solutions, we substitute each value of x back into the original equation and check if the equation holds true.
Verifying x = 12
Substituting x = 12 into the equation 3(x - 4)^2 = 192, we get:
3(12 - 4)^2 = 3(8)^2 = 3(64) = 192
The equation holds true, confirming that x = 12 is indeed a solution.
Verifying x = -4
Now, let us substitute x = -4 into the same equation:
3(-4 - 4)^2 = 3(-8)^2 = 3(64) = 192
Again, the equation holds true, verifying that x = -4 is also a solution.
With both solutions verified, we can confidently conclude that the numbers that satisfy the given condition are 12 and -4.
The Answer
The numbers are 12, -4.
The Significance of Mathematical Problem-Solving
The journey we have undertaken to solve this mathematical problem is not merely an exercise in number manipulation. It is a testament to the power of mathematical thinking, a process that involves breaking down complex problems into manageable steps, applying logical reasoning, and arriving at precise solutions. The skills we have honed in this process are not confined to the realm of mathematics alone; they are transferable to various aspects of life, empowering us to approach challenges with clarity, confidence, and a systematic mindset.
The Beauty of Mathematical Exploration
Mathematics is not just a collection of formulas and equations; it is a vibrant and dynamic field of exploration, a realm where we can discover patterns, uncover relationships, and unravel the mysteries of the universe. Each problem we solve, each theorem we prove, is a step forward in our understanding of the world around us. The satisfaction derived from solving a challenging mathematical problem is akin to the joy of completing a complex puzzle, a feeling of accomplishment that fuels our intellectual curiosity and inspires us to delve deeper into the world of mathematics.
The Importance of Persistence and Perseverance
Mathematical problem-solving is not always a smooth and linear process. There will be times when we encounter roadblocks, when the solution seems elusive, and when we feel tempted to give up. It is in these moments that persistence and perseverance become our greatest allies. By refusing to be discouraged by setbacks, by approaching the problem from different angles, and by seeking inspiration from others, we can overcome obstacles and ultimately achieve our goals. The ability to persevere in the face of challenges is a valuable life skill that is cultivated through mathematical problem-solving.
The Collaborative Nature of Mathematics
While mathematical problem-solving often involves individual effort, it is also a collaborative endeavor. Mathematicians throughout history have built upon the work of their predecessors, sharing ideas, challenging assumptions, and collectively advancing our understanding of the mathematical world. Today, mathematicians continue to collaborate across institutions and countries, working together to solve the most challenging problems facing humanity. The collaborative nature of mathematics fosters creativity, innovation, and a sense of community among those who share a passion for the subject.
Conclusion A Journey of Discovery
Our journey to find the number(s) that satisfy the condition three times the square of the difference between a number and 4 is 192 has been a rewarding experience. We have successfully translated a verbal problem into a mathematical equation, solved the equation using algebraic techniques, verified our solutions, and reflected on the significance of mathematical problem-solving. This journey has not only enhanced our mathematical skills but has also provided us with valuable insights into the nature of problem-solving itself.
The numbers we found, 12 and -4, are not merely numerical answers; they are the fruits of our intellectual labor, the tangible results of our perseverance and dedication. As we continue our exploration of the mathematical world, let us carry with us the lessons we have learned from this journey, embracing the challenges, celebrating the discoveries, and cherishing the beauty of mathematical thinking.