Solving The Square Of 9 Less Than A Number Problem

Mathematics often presents us with intriguing puzzles that challenge our problem-solving skills. In this article, we will delve into one such puzzle, a classic word problem that involves translating verbal statements into algebraic equations and then solving those equations to find the unknown number. The problem statement is: "The square of 9 less than a number is 3 less than the number. What is the number?" This seemingly simple statement holds a wealth of mathematical concepts, including variables, algebraic expressions, quadratic equations, and the crucial skill of interpreting word problems. Let's embark on a journey to dissect this problem, understand its underlying structure, and ultimately arrive at the correct solution. This problem isn't just about finding a number; it's about the process of mathematical reasoning and the ability to translate real-world scenarios into mathematical models. We'll explore how to represent the unknown number with a variable, form an equation based on the given information, and then employ algebraic techniques to solve for that variable. By the end of this exploration, you'll not only have the answer but also a deeper understanding of how to approach similar mathematical challenges. Understanding the relationship between words and mathematical symbols is a fundamental skill in algebra. It allows us to take complex scenarios and break them down into manageable equations that we can then solve. This problem highlights the importance of careful reading and attention to detail, as even a slight misinterpretation can lead to an incorrect equation and, consequently, an incorrect solution. So, let's dive in and unravel the mystery behind this intriguing mathematical puzzle.

H2: Translating Words into Mathematical Expressions

The first step in tackling this problem is to translate the words into mathematical expressions. This is a crucial step in solving any word problem, as it allows us to move from the realm of verbal statements to the precise language of algebra. Let's break down the problem statement piece by piece. The phrase "a number" is our unknown quantity, and in algebra, we represent unknowns with variables. So, let's assign the variable x to represent this unknown number. Now, let's consider the phrase "9 less than a number." This means we are subtracting 9 from our number x. So, we can represent this as x - 9. The next key phrase is "the square of 9 less than a number." This means we need to take the expression x - 9 and square it, which gives us (x - 9)^2. Now, let's move on to the second part of the statement: "3 less than the number." This means we are subtracting 3 from our original number x, resulting in the expression x - 3. Finally, the problem states that "the square of 9 less than a number is 3 less than the number." This gives us the crucial equation that connects the two expressions we've derived. We can write this equation as (x - 9)^2 = x - 3. This equation is the heart of our problem, and solving it will lead us to the solution. By carefully translating each phrase into mathematical symbols, we've successfully transformed the word problem into an algebraic equation. This is a fundamental skill in algebra, and mastering it opens the door to solving a wide range of mathematical problems. The ability to translate words into mathematical expressions is not just about manipulating symbols; it's about understanding the underlying relationships and representing them accurately. It requires a careful reading of the problem statement, identifying the key quantities and operations, and then expressing those relationships using mathematical notation. With practice, this skill becomes second nature, allowing you to approach word problems with confidence and clarity.

H2: Forming the Equation and Solving for x

Now that we have translated the word problem into the algebraic equation (x - 9)^2 = x - 3, the next step is to solve for the unknown variable x. This involves expanding the squared term, simplifying the equation, and then using algebraic techniques to isolate x. Let's start by expanding the left side of the equation. (x - 9)^2 is equivalent to (x - 9)(x - 9). Using the distributive property (or the FOIL method), we get x^2 - 9x - 9x + 81, which simplifies to x^2 - 18x + 81. Now our equation looks like this: x^2 - 18x + 81 = x - 3. To solve for x, we need to rearrange the equation so that all the terms are on one side and the equation is set to zero. This is because we recognize that this is a quadratic equation, and quadratic equations are typically solved by setting them equal to zero. Subtracting x from both sides gives us x^2 - 19x + 81 = -3. Adding 3 to both sides gives us the standard form of the quadratic equation: x^2 - 19x + 84 = 0. Now we need to solve this quadratic equation. There are several methods we can use, including factoring, completing the square, or using the quadratic formula. In this case, factoring is a relatively straightforward approach. We are looking for two numbers that multiply to 84 and add up to -19. By considering the factors of 84, we find that -7 and -12 satisfy these conditions, since (-7) * (-12) = 84 and (-7) + (-12) = -19. Therefore, we can factor the quadratic equation as (x - 7)(x - 12) = 0. For the product of two factors to be zero, at least one of the factors must be zero. This gives us two possible solutions: x - 7 = 0 or x - 12 = 0. Solving these linear equations, we get x = 7 or x = 12. These are the two solutions to our equation, and they represent the numbers that satisfy the original problem statement. By carefully expanding, simplifying, and factoring, we've successfully solved the quadratic equation and found the values of x that make the equation true. This process highlights the importance of algebraic manipulation and the ability to recognize and apply different solution techniques.

H2: Verifying the Solutions

Having obtained two potential solutions, x = 7 and x = 12, it's crucial to verify whether these solutions indeed satisfy the original problem statement. This step is essential in problem-solving, as it helps to catch any potential errors in our calculations or reasoning. It's also a good practice to ensure that our solutions make sense in the context of the problem. Let's start by verifying x = 7. According to the problem statement, the square of 9 less than the number should be 3 less than the number. If x = 7, then 9 less than the number is 7 - 9 = -2. The square of -2 is (-2)^2 = 4. Now, 3 less than the number is 7 - 3 = 4. Since 4 = 4, the solution x = 7 satisfies the original problem statement. Next, let's verify x = 12. If x = 12, then 9 less than the number is 12 - 9 = 3. The square of 3 is 3^2 = 9. Now, 3 less than the number is 12 - 3 = 9. Since 9 = 9, the solution x = 12 also satisfies the original problem statement. Therefore, both x = 7 and x = 12 are valid solutions to the problem. By verifying our solutions, we gain confidence in our answer and ensure that we have not made any mistakes in our calculations or reasoning. This step also reinforces the importance of connecting the algebraic solutions back to the original problem context. Verification is a critical part of the problem-solving process, especially in mathematics, where precision and accuracy are paramount. It's a way to double-check our work and ensure that our solutions are not only mathematically correct but also logically sound within the context of the problem. This practice enhances our understanding of the problem and strengthens our problem-solving skills.

H2: The Answer and its Significance

After meticulously translating the word problem into an algebraic equation, solving the equation, and verifying the solutions, we have arrived at the answer. The numbers that satisfy the condition "the square of 9 less than a number is 3 less than the number" are 7 and 12. This corresponds to option A in the given choices. The significance of this problem extends beyond simply finding the correct answer. It highlights the power of algebra as a tool for representing and solving real-world problems. By translating verbal statements into mathematical expressions, we can transform complex scenarios into manageable equations that can be solved using algebraic techniques. This problem also demonstrates the importance of careful attention to detail. A slight misinterpretation of the problem statement can lead to an incorrect equation and, consequently, an incorrect solution. The ability to break down a complex problem into smaller, more manageable steps is a crucial skill in mathematics and in life. This problem showcases this skill, as we systematically translated the words into expressions, formed the equation, solved it, and then verified the solutions. Furthermore, this problem illustrates the concept of quadratic equations and their solutions. Quadratic equations often have two solutions, as we saw in this case. Understanding this concept is essential for solving a wide range of mathematical problems. In conclusion, the problem of finding the numbers that satisfy the given condition is not just about the answer; it's about the journey of problem-solving. It's about the skills we develop along the way, including translation, equation solving, and verification. These skills are valuable not only in mathematics but also in other areas of life where problem-solving is essential. The solutions 7 and 12 represent the culmination of this process, a testament to the power of mathematical reasoning and the importance of a systematic approach.

H2: Conclusion: Mastering the Art of Problem-Solving

In summary, we have successfully navigated the challenge of determining the number whose square, when 9 is subtracted, results in a value that is 3 less than the original number. Through a meticulous process of translating words into algebraic expressions, constructing and solving a quadratic equation, and verifying the solutions, we arrived at the answers: 7 and 12. This exercise underscores the significance of mathematical literacy in deciphering and resolving real-world scenarios. The ability to convert verbal statements into mathematical equations is a cornerstone of problem-solving, applicable far beyond the classroom. The problem further exemplifies the multifaceted nature of algebra, showcasing the necessity of skills ranging from basic arithmetic operations to the manipulation of algebraic expressions and the application of factoring techniques. The presence of two solutions highlights the richness of quadratic equations and the importance of considering all possible outcomes. Beyond the specific solution, this problem serves as a paradigm for effective problem-solving strategies. It emphasizes the value of breaking down complex problems into smaller, manageable steps, a skill that is transferable to various domains. The emphasis on verification reinforces the importance of accuracy and attention to detail in mathematical endeavors. Ultimately, this exploration is not just about finding the answer to a single mathematical problem. It is about cultivating a mindset of critical thinking, logical reasoning, and systematic problem-solving. These are skills that are not only essential for success in mathematics but also for navigating the challenges and opportunities of everyday life. By mastering the art of problem-solving, we empower ourselves to approach new situations with confidence and to transform challenges into opportunities for growth and learning. The journey of solving this problem has provided valuable insights into the power of mathematics and the importance of a structured approach to problem-solving, skills that will serve us well in our academic pursuits and beyond.