The Sum Of A Number And Its Square Is 42. Which Equation Can Represent This Situation To Find The Number?

In the realm of mathematics, word problems often present intriguing challenges that require us to translate verbal statements into algebraic equations. These equations then serve as the foundation for solving for unknown quantities. One such problem involves finding a number whose sum with its square equals 42. This seemingly simple problem delves into the heart of quadratic equations, a fundamental concept in algebra. Let's embark on a journey to decipher this mathematical puzzle, explore the concepts involved, and ultimately arrive at the correct equation that can unravel the mystery numbers. Understanding the relationship between words and equations is key to mathematical problem-solving. The ability to accurately translate a word problem into a mathematical equation is a crucial skill that forms the bedrock of algebraic reasoning. This process involves identifying the unknown quantities, representing them with variables, and then expressing the given relationships using mathematical symbols and operations. In this particular problem, we are presented with a scenario where the sum of a number and its square is equal to 42. Our mission is to convert this verbal description into a concise and precise algebraic equation that captures the essence of the problem. As we delve into the intricacies of this problem, we will encounter the concept of quadratic equations, which play a significant role in various mathematical and scientific applications. A quadratic equation is a polynomial equation of the second degree, characterized by the presence of a term where the variable is raised to the power of 2. These equations often arise in situations involving areas, projectile motion, and other real-world scenarios. By understanding the structure and properties of quadratic equations, we can effectively solve them and extract valuable information about the problem at hand. This problem serves as a stepping stone to more complex mathematical concepts, highlighting the importance of mastering the fundamentals. It underscores the power of algebra in translating real-world scenarios into mathematical models that can be analyzed and solved. By successfully navigating this problem, we not only hone our equation-solving skills but also gain a deeper appreciation for the elegance and applicability of mathematics in our daily lives.

Translating Words into Algebra: The Key to the Solution

To solve this problem effectively, the first critical step involves translating the given word problem into a mathematical equation. This process requires careful attention to the wording and identifying the key relationships between the unknown quantities. Let's break down the problem statement step by step: "The sum of a number and its square is 42." The phrase "a number" immediately suggests that we have an unknown quantity, which we can represent using a variable. Let's choose the variable 'x' to represent this number. The next crucial part of the statement is "its square." In mathematical terms, the square of a number means the number multiplied by itself. Therefore, the square of 'x' can be written as x². Now, we encounter the phrase "the sum of." This indicates that we need to add two quantities together. In our case, we are adding the number 'x' and its square, x². Finally, the phrase "is 42" tells us that the sum we just calculated is equal to 42. This gives us the foundation for constructing our equation. Combining all these elements, we can express the given statement as an algebraic equation: x² + x = 42. This equation accurately captures the relationship described in the word problem. It states that the sum of the number 'x' and its square x² is equal to 42. This equation is a quadratic equation because it involves a term with x raised to the power of 2. Quadratic equations have a distinctive form and can be solved using various techniques, such as factoring, completing the square, or using the quadratic formula. Now that we have successfully translated the word problem into an algebraic equation, we have taken a significant step towards finding the solution. The equation x² + x = 42 serves as a mathematical representation of the problem, allowing us to apply algebraic methods to determine the values of 'x' that satisfy the given condition. This process of translating words into equations is a fundamental skill in algebra, enabling us to solve a wide range of problems in mathematics and other disciplines. This ability to bridge the gap between verbal descriptions and mathematical expressions is what empowers us to analyze and solve real-world problems using the tools of algebra.

Analyzing the Options: Identifying the Correct Equation

Now that we have derived the equation x² + x = 42 from the problem statement, let's examine the provided options and determine which one matches our derived equation. This step involves comparing our equation with the given choices and identifying the exact match. Option A states: x² + x = 42. This equation is precisely the same as the equation we derived from the problem statement. It accurately represents the relationship between the number and its square, stating that their sum is equal to 42. Therefore, Option A is a strong contender for the correct answer. Option B presents the equation: x² + 2x = 42. This equation differs from our derived equation in the term involving 'x'. Instead of 'x', this equation has '2x', which means twice the number. This equation does not accurately represent the original problem statement, as it introduces an additional factor of 2 that is not present in the given information. Therefore, Option B is not the correct equation. Option C provides the equation: x² + x + 42 = 0. While this equation contains the terms x² and x, it also includes an additional term of +42 on the same side of the equation as the x² and x terms. This equation is not equivalent to our derived equation, which has 42 on the opposite side of the equation. Option C represents a different mathematical relationship and is therefore not the correct equation for this problem. Option D presents the equation: x² + 2x + 42 = 0. Similar to Option C, this equation includes the terms x² and 2x, as well as a constant term of +42 on the same side of the equation. This equation deviates from our derived equation and does not accurately represent the original problem statement. Therefore, Option D is not the correct equation. By carefully comparing our derived equation with the provided options, we can confidently conclude that Option A, x² + x = 42, is the correct equation that can be used to find the two numbers for which the sum of the number and its square is 42. This process of analyzing options and comparing them with a derived result is a crucial skill in problem-solving, particularly in mathematics. It allows us to verify our work, eliminate incorrect choices, and arrive at the accurate solution.

The Correct Equation: A Foundation for Solving the Problem

After careful analysis, we have identified Option A, x² + x = 42, as the correct equation that accurately represents the given word problem. This equation serves as the foundation for solving for the unknown number. Understanding why this equation is correct is crucial for grasping the underlying mathematical concept. The equation x² + x = 42 captures the essence of the problem statement: "The sum of a number and its square is 42." The left-hand side of the equation, x² + x, represents the sum of the square of the number (x²) and the number itself (x). The right-hand side of the equation, 42, represents the value that this sum is equal to. By setting these two expressions equal to each other, we create an algebraic equation that can be manipulated and solved to find the values of 'x' that satisfy the given condition. This equation is a quadratic equation, which means it has a term where the variable 'x' is raised to the power of 2. Quadratic equations have a distinctive form and can have up to two solutions. Solving this equation will provide us with the two numbers that satisfy the condition that their sum with their squares is equal to 42. The fact that the equation is quadratic implies that there might be two possible solutions for 'x'. This is because quadratic equations often have two roots, which are the values of the variable that make the equation true. In the context of this problem, it means that there could be two different numbers that, when added to their squares, result in 42. To find these solutions, we can employ various techniques for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. Each of these methods provides a systematic approach to isolating the variable 'x' and determining its possible values. The correct equation, x² + x = 42, is not just a symbolic representation of the problem; it is a powerful tool that enables us to find the specific numbers that meet the given criteria. By understanding the structure and properties of this equation, we can unlock the solution and gain a deeper understanding of the relationship between numbers and their squares.

Beyond the Equation: The Power of Mathematical Translation

Identifying the correct equation, x² + x = 42, is a significant step, but the true power lies in understanding the process of mathematical translation that led us to this equation. This problem highlights the fundamental skill of converting verbal descriptions into symbolic representations, a cornerstone of mathematical problem-solving. The ability to translate words into equations is not merely a mechanical exercise; it is a process of critical thinking and interpretation. It requires us to carefully analyze the given information, identify the key relationships, and express them using mathematical symbols and operations. This skill is not limited to algebra; it extends to various branches of mathematics and other scientific disciplines. In physics, for example, we often need to translate real-world phenomena into mathematical models to analyze and predict their behavior. In economics, we use equations to represent economic relationships and make forecasts. In computer science, algorithms are expressed using mathematical notation. The process of mathematical translation involves several key steps. First, we must identify the unknown quantities and represent them with variables. This step requires us to understand what the problem is asking us to find. Next, we need to identify the relationships between these quantities. This often involves looking for keywords such as "sum," "difference," "product," "quotient," and "is equal to." These keywords provide clues about the mathematical operations that need to be used. Finally, we combine the variables and operations to form an equation or a system of equations. This equation or system represents the mathematical model of the problem. This problem serves as an excellent example of how mathematical translation can be used to solve real-world problems. By converting the verbal description into an algebraic equation, we were able to apply the tools of algebra to find the solution. This process not only provides the answer but also enhances our understanding of the underlying mathematical concepts. The ability to translate words into equations is a valuable skill that empowers us to tackle a wide range of problems in mathematics and beyond. It is a skill that is honed through practice and a deep understanding of mathematical principles.

In conclusion, the problem of finding the equation that represents the sum of a number and its square being 42 underscores the importance of mastering the art of equation formulation. We successfully identified Option A, x² + x = 42, as the correct equation by carefully translating the verbal statement into an algebraic expression. This process involved recognizing the unknown quantity, representing it with a variable, and then expressing the given relationship using mathematical symbols and operations. This problem serves as a valuable lesson in the power of algebra to model real-world scenarios. By translating words into equations, we can unlock solutions to problems that might otherwise seem intractable. The ability to formulate equations is a fundamental skill in mathematics and is essential for success in various fields, including science, engineering, and finance. Furthermore, this problem highlights the significance of understanding quadratic equations. The equation x² + x = 42 is a quadratic equation, which means it has a term where the variable is raised to the power of 2. Quadratic equations have distinctive properties and can be solved using various techniques. Mastering the techniques for solving quadratic equations is crucial for tackling a wide range of mathematical problems. Moreover, the problem-solving process we employed in this case underscores the importance of careful analysis and attention to detail. We meticulously broke down the problem statement, identified the key relationships, and then translated them into an algebraic equation. We also carefully analyzed the provided options, comparing them with our derived equation to identify the correct match. This systematic approach is essential for success in any mathematical problem-solving endeavor. In summary, the problem of finding the equation that represents the sum of a number and its square being 42 provides valuable insights into the power of mathematical translation, the importance of understanding quadratic equations, and the significance of careful analysis in problem-solving. By mastering these skills, we can confidently tackle a wide range of mathematical challenges and unlock the beauty and applicability of mathematics in our daily lives. The journey from words to equations is a testament to the elegance and power of mathematical thinking.