Equation For Sum Of Number And Its Square Equals 42

by Admin 52 views
Equation for Sum of Number and Its Square Equals 42

Let's dive into a classic math problem! We're going to break down how to form an equation when the sum of a number and its square equals 42. It sounds tricky, but trust me, it's totally doable. We'll go step-by-step so you can conquer similar problems in the future. So, if you've ever scratched your head at word problems, especially those involving quadratics, you're in the right place. We'll make it crystal clear! Stick around, and let's get this math party started!

Understanding the Problem

Okay, guys, first things first, let's really understand what the problem is asking. When you see a word problem, it's like reading a mini-story, but instead of characters and plot twists, we have numbers and relationships. Our mission? Translate those words into math! In this case, we have a number, its square, and a sum that equals 42. Those are our key ingredients. Think of it like a recipe – we just need to put them together in the right order. The most important part of this step is identifying the unknowns and the relationships between them. Once we have a clear picture in our minds, the rest becomes much easier. So, let’s put on our detective hats and zoom in on those keywords!

Breaking Down the Keywords

Let's dissect the keywords to make sure we're all on the same page. The phrase "a number" immediately suggests we're dealing with an unknown. In math, we love using variables (like x, y, or even a smiley face if you're feeling wild!), to represent unknowns. So, let's call our mystery number x. Next up, "its square." When we square a number, we're simply multiplying it by itself. So, the square of x is x times x, which we write as x². See? We're already translating! Now, we have "the sum." Sum means addition – we're adding two things together. And finally, "equals 42" is pretty straightforward – it means our final result is 42. Armed with these keyword translations, we're ready to start constructing our equation. It's like having the pieces of a puzzle; now, we just need to fit them together.

Translating Words into Math

Now comes the fun part: turning those words into a mathematical equation! We know "a number" is x, "its square" is x², and "the sum" means we're adding them. So, we can write x + x². The problem tells us this sum "is" 42. In math language, "is" often means equals (=). So, we can complete our equation: x + x² = 42. But wait, there's a bit of math etiquette we need to follow. It's standard practice to write polynomial equations (that's what we have here!) in descending order of exponents. That means we want the x² term first, then the x term, and finally the constant term. So, let's rearrange our equation to x² + x = 42. We're almost there – just one more little tweak to make it look like the answer choices we often see in tests.

Forming the Equation

Alright, let’s nail down the final form of our equation. We've got x² + x = 42, which is excellent progress. However, most standard quadratic equations are set equal to zero. This makes them easier to solve using various methods like factoring, completing the square, or the quadratic formula (we'll save those for another day!). So, how do we get our equation equal to zero? Simple! We subtract 42 from both sides. Remember, in math, whatever you do to one side of the equation, you must do to the other to keep things balanced. So, subtracting 42 from both sides of x² + x = 42 gives us x² + x - 42 = 0. Boom! We've transformed our equation into the standard quadratic form. This is the key to unlocking the solutions for x, which are the numbers that satisfy the original problem. Now, let's take a look at how this lines up with the options we're given.

Step-by-Step Breakdown

Let's recap the steps we took to build our equation. This is super helpful for tackling similar problems in the future.

  1. Identify the Unknown: We recognized "a number" as our unknown and assigned it the variable x.
  2. Express the Square: We translated "its square" into x².
  3. Form the Sum: We added the number and its square: x + x².
  4. Set up the Equality: We equated the sum to 42: x + x² = 42.
  5. Rearrange (Optional): We rearranged the terms in descending order of exponents: x² + x = 42.
  6. Set Equal to Zero: We subtracted 42 from both sides to get the standard quadratic form: x² + x - 42 = 0.

By following these steps, you can confidently translate word problems into mathematical equations. It's like having a recipe for success! Practice makes perfect, so the more you break down problems like this, the easier it will become. Now, let’s see how this compares to the answer choices.

Analyzing the Answer Choices

Now, let's play the comparison game. We've arrived at our equation: x² + x - 42 = 0. It's time to see which of the given answer choices matches our hard work. This is a crucial step because sometimes answer choices can be sneaky! They might try to trick you with slight variations or rearrangements of the equation. So, we need to be extra careful and pay close attention to the signs and coefficients. Think of it like a matching puzzle – we're looking for the piece that fits perfectly. By carefully comparing our equation to the options, we can confidently select the correct answer and avoid any potential traps.

Comparing Our Equation

Time to put on our detective hats again and scrutinize those answer choices. We're looking for an equation that looks exactly like ours: x² + x - 42 = 0.

  • Choice A looks promising, but let's double-check every term and sign.
  • Choices B, C, and D might have some similarities, but we need to be sure they match our equation perfectly.

Remember, even a tiny difference can make an answer choice incorrect. It's like baking a cake – if you add too much of one ingredient, the whole thing can be ruined! So, let's be meticulous and ensure we choose the option that is the spitting image of our equation. This careful comparison will lead us to the correct answer with confidence.

Identifying the Correct Option

After carefully comparing our equation, x² + x - 42 = 0, with the answer choices, we can pinpoint the winner. Option A, x² + x = 42, looks like a strong contender. However, it's not quite in the same form as our equation, which is set equal to zero. To make sure, let’s do a quick transformation. If we subtract 42 from both sides of Option A, we get x² + x - 42 = 0. Bingo! It's a match! This confirms that Option A correctly represents the relationship described in the problem. We’ve solved it, guys! By systematically breaking down the problem, translating the words into math, and carefully comparing our result with the answer choices, we've nailed it. This is how you conquer math problems – step by step, with confidence and a dash of detective work!

Conclusion

So there you have it, folks! We've successfully transformed a word problem into a mathematical equation. Remember, the key is to break it down step-by-step, translate those keywords, and don't be afraid to rearrange things until they look familiar. Math problems can seem daunting at first, but with a little practice and the right approach, you can totally crush them. Now you're equipped to tackle similar problems with confidence. Keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!