Finding The Number: A Math Puzzle Explained

Hey math enthusiasts! Let's dive into a classic problem: "The square of a number is 12 less than 7 times the number. What is the number?" This isn't just a math problem; it's a chance to flex those problem-solving muscles and understand how algebra works in the real world. Think of it like a treasure hunt where we have to uncover the hidden value. We're going to break down this question step-by-step, making sure that it is easy to understand. We'll start with the basics, create an equation, solve it, and get to the correct answers. So, grab your pencils, and let's get started, guys!

First off, let's break down the problem statement. The core of this question lies in translating words into mathematical language. The phrase "the square of a number" immediately tells us that we're dealing with an unknown value (let's call it 'x') that's been multiplied by itself (x * x, which is written as x²). The key here is not to get confused by the wording but to focus on the facts. The whole point is to represent everything mathematically. Next, we have "12 less than 7 times the number." This part tells us about the result of our calculation. "7 times the number" is written as 7x and "12 less than" means we subtract 12 from this product. Therefore, it is 7x - 12. And the crucial link: "the square of a number is" means that the results of both parts are equal. Therefore, we understand how to make the equation.

Now, let's turn this into an equation. From the problem statement, we know that the square of a number (x²) equals 12 less than 7 times that number (7x - 12). Therefore, we have the equation: x² = 7x - 12. This is what we call a quadratic equation. To solve it, we need to rearrange it so that one side of the equation equals zero. We do this by subtracting 7x and adding 12 from both sides, which gives us: x² - 7x + 12 = 0. This is the standard form of a quadratic equation, which makes it easier to solve using factoring, completing the square, or the quadratic formula. At this point, you can choose your favorite method. But for this specific equation, factoring is probably the easiest route to take.

So how to tackle it? The strategy here is to find two numbers that multiply to give us 12 (the constant term) and add up to -7 (the coefficient of the x term). If you think about it, the numbers -3 and -4 fit the bill. Because (-3) * (-4) = 12 and (-3) + (-4) = -7. Using these numbers, we can factor the quadratic equation. So we can rewrite the equation x² - 7x + 12 = 0 as (x - 3)(x - 4) = 0. We've simplified the equation into a product of two binomials, each containing our variable, 'x'. The goal here is to isolate x. For the whole equation to equal zero, one of the factors must equal zero. Think of it like this: anything multiplied by zero is zero. Therefore, either (x - 3) = 0 or (x - 4) = 0.

Decoding the Equation: Finding the Right Answers

Once we have our factored equation, (x - 3)(x - 4) = 0, we're very close to the solution. The core concept here is understanding the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This lets us break down the problem into two simpler equations. Solving each one will give us the two possible values of 'x' that satisfy the original equation. Let’s break it down further.

So, if (x - 3) = 0, we can solve for x by adding 3 to both sides, which gives us x = 3. Similarly, if (x - 4) = 0, we can add 4 to both sides of the equation to get x = 4. The solutions to our original problem are x = 3 and x = 4. These are the two numbers whose square is 12 less than 7 times the number. It's like finding the two keys that unlock the treasure chest. But wait, we're not quite done. It's always a good idea to check our answers! Doing this is called verifying our solutions. That means making sure that the values we found actually work in the original equation. Let's start with x = 3. If x = 3, then x² = 9 and 7x - 12 = (7 * 3) - 12 = 21 - 12 = 9. Bingo! Both sides of the equation are equal, which confirms that 3 is indeed a solution.

Now, let’s test x = 4. If x = 4, then x² = 16 and 7x - 12 = (7 * 4) - 12 = 28 - 12 = 16. Another success! Both sides of the equation are equal, which confirms that 4 is a solution. This is really about understanding and working with quadratic equations.

So, to sum it up: x = 3 and x = 4 are the values that make the statement true. This means that if we square 3, we get 9, and if we take 7 times 3 and subtract 12 (73 - 12), we also get 9. The same logic works for the number 4: 4 squared is 16, and 7 times 4 minus 12 (74 - 12) is also 16. The most crucial part of this problem is to translate words into a mathematical equation and, then, solve it.

The Correct Answer and Why Others Are Incorrect

Now that we've found our solutions, let's revisit the answer choices to see which one matches our findings. The correct answer is D: 3 or 4. This option accurately reflects the two possible values of the number that satisfy the original problem. The other options, A, B, and C, are incorrect because they include either incorrect values or a mix of positive and negative numbers.

Let's break down why these are wrong: Option A suggests -4 or -3. If we put -4 into our original equation, the square of -4 is 16, and 7 times -4 minus 12 is -40, so it’s incorrect. The same thing can be said for -3. Therefore, those values don't satisfy the original conditions of the problem. Option B has -4 or 3. As we have seen, the -4 is incorrect. Option C has -3 or 4. Again, as we have seen, the -3 is incorrect. The point is to understand that the problem is not about just finding any two numbers, but two specific numbers that satisfy the core equation. Therefore, always verify. This step is the most critical for solving math problems. This process ensures that the solutions are logically sound. Therefore, it is important to choose the answer that correctly identifies all the valid solutions.

So remember, the next time you face a word problem, don't be intimidated! Take it step-by-step, translate the words into an equation, solve the equation using the appropriate methods, and always verify your answers. You've got this!

This kind of problem helps build a solid foundation in algebra. It helps you practice how to translate real-world scenarios into mathematical equations, and it helps you understand how to solve equations.

Key Takeaways: Mastering the Math Puzzle

Let’s summarize the key takeaways from this problem, guys!

• Translation is Key: The most important step is to translate the words of a problem into a correct mathematical equation. Carefully read the problem and write down what you know.
• Quadratic Equations: Recognize that this type of problem involves a quadratic equation, which means it will likely have two solutions. That's why we end up with two possible numbers.
• Factoring and Solving: Knowing how to factor quadratic equations is a very useful skill. It's not the only method, but it's a fast way to get to the answer. Remember to use the zero-product property after factoring to get your answers.
• Verification: Always double-check your answers by plugging them back into the original equation. This is the easiest way to make sure that your solutions are correct and to avoid making silly mistakes.
• Practice Makes Perfect: The more problems you solve, the more comfortable you'll become with these kinds of questions. Practice regularly, and don't be afraid to try different methods.

This problem is a solid example of the link between the real world and mathematical ideas. It demonstrates how algebra can be used to solve real-world problems. Keep practicing, keep learning, and don't hesitate to ask for help whenever you need it. Math is a journey, and every problem is a step toward understanding the world around us. So, keep up the fantastic work and embrace the fun of solving math puzzles! Good luck with your math studies! And remember, practice makes perfect!