Finding Numbers: A Math Problem Solved!

Hey math enthusiasts! Let's dive into a classic problem that combines algebra and a bit of critical thinking. We're going to tackle a problem where one number is five more than another, and we'll use equations to figure out what those numbers actually are. This is a great exercise to sharpen your problem-solving skills and get comfortable with algebraic manipulations. Ready to get started? Let's break it down step by step and make sure you grasp all the concepts involved. We'll start with the basics, define our variables, set up the equations, and finally, find the solution. Don't worry if it seems daunting at first; we'll go at a pace that keeps everything clear. This journey will highlight how math is used in everyday scenarios and how it helps us understand the world around us. So, whether you're a student prepping for a test, or just curious about math puzzles, stick around. You will not only solve the problem, but also boost your confidence in solving similar challenges.

Okay, here's the problem: One number is 5 greater than another number. If the sum of their squares is five times the square of the smaller number, what are the numbers? Let's start by understanding what the problem is asking. We need to find two numbers. We know there's a relationship between them: one is larger than the other by 5. We also know something about their squares. The sum of the squares of these numbers has a special relationship to the square of the smaller number. To solve this, we're going to translate the words into equations. This is where we bring in the magic of algebra. Once we have the equations, we can solve them. Trust me, it's not as scary as it sounds. Let's start with some definitions, because that's always a good starting point. This foundation will help make the problem much clearer and allow us to identify all of the components that we will need in order to reach the final answer. We will carefully define what we already know and what we are trying to find, step-by-step.

Setting Up the Problem: Defining Variables and Equations

Alright, guys, let's get our hands dirty with some variables. In algebra, variables are like placeholders. We use them to represent numbers we don't know yet. This is super helpful because it allows us to create relationships that we can then use to solve for the unknown. First things first, let's define our variables. Let's call the smaller number 'x'. Since the other number is 5 greater than the smaller number, we can represent that number as 'x + 5'. See how we've already started translating the words into mathematical expressions? Pretty neat, right? Now, let's translate the second part of the problem. It says, 'the sum of their squares is five times the square of the smaller number.' Here’s how we'll write that as an equation. The square of the smaller number is x². The square of the larger number is (x + 5)². The sum of their squares is x² + (x + 5)². And finally, this sum is equal to five times the square of the smaller number, which is 5x². Therefore, our equation becomes: x² + (x + 5)² = 5x². This is the core of our problem, and everything will revolve around this equation. Understanding how we got here is crucial. We will take this step-by-step, making sure that we don't get lost in the algebra. It is important to note that the correct setup of equations is as important as the solving stage, because if we make a mistake during the setup process, then the answer will not be correct.

Now, let's revisit each of the pieces of information to ensure that we did not miss anything important. One number is 5 greater than another number, and so we decided that x is the smaller number and x + 5 is the larger number. This relationship is a fundamental one, and we must make sure that we have it clear in our minds before proceeding. The second piece of information tells us something about the squares of these numbers: the sum of their squares is five times the square of the smaller number. The sum of the squares is written as x² + (x + 5)², and the square of the smaller number is written as x², which, when multiplied by 5, results in 5x². We then know that we will be working with an equation of the form x² + (x + 5)² = 5x², and this is the entire setup of our problem. The next part will be the solving stage.

Solving the Equation: Step-by-Step Guide

Okay, guys, time to roll up our sleeves and solve the equation! We have our equation: x² + (x + 5)² = 5x². This might look a little intimidating at first, but trust me, it's manageable. Our goal is to isolate 'x' and find its value. First, let's expand the (x + 5)² part. Remember, (x + 5)² means (x + 5) times (x + 5). When we multiply that out, we get x² + 10x + 25. Now, our equation looks like this: x² + x² + 10x + 25 = 5x². Combine like terms on the left side, which gives us 2x² + 10x + 25 = 5x². Next, we want to bring everything to one side to set the equation to zero. Subtract 5x² from both sides to get 2x² + 10x + 25 - 5x² = 0. Simplify that, and we get -3x² + 10x + 25 = 0. We can multiply the entire equation by -1 to make the leading coefficient positive. So, our equation becomes 3x² - 10x - 25 = 0. Now, we have a quadratic equation. We can solve this by factoring, using the quadratic formula, or completing the square. Let's try factoring. We're looking for two numbers that multiply to give us -75 (3 times -25) and add up to -10. Those numbers are -15 and 5. So we can rewrite the middle term as -15x + 5x: 3x² - 15x + 5x - 25 = 0. Now factor by grouping. From the first two terms, we can factor out 3x, giving us 3x(x - 5). From the last two terms, we can factor out 5, giving us 5(x - 5). So our equation is now 3x(x - 5) + 5(x - 5) = 0. Factor out the common term (x - 5), and we get (3x + 5)(x - 5) = 0. For this equation to be true, either (3x + 5) = 0 or (x - 5) = 0. Solving these, we get x = -5/3 or x = 5. These are the possible values for 'x'.

We started with a quadratic equation, which is an equation where the highest power of the variable is two. We transformed the equation, combined like terms, and set the equation to zero, which gave us the format ax² + bx + c = 0. We then used the factoring technique to arrive at the values of x. Factoring is a handy way to solve quadratic equations, where we try to break down a quadratic expression into the product of two simpler expressions (usually binomials). This method is based on the idea of reversing the process of multiplying binomials. When we factor, we're looking for two binomials that, when multiplied together, produce the original quadratic expression. The key to factoring is finding the right combination of numbers and signs that satisfy the relationships between the coefficients and the constant term of the quadratic equation. The quadratic formula is another very important method when solving these types of equations. This formula provides a direct way to find the roots of any quadratic equation of the form ax² + bx + c = 0. The quadratic formula is: x = (-b ± √(b² - 4ac)) / (2a). By substituting the values of a, b, and c into this formula, we can calculate the values of x. Understanding quadratic equations is really important, as they come up again and again in mathematics and in real life. They model various phenomena in physics, engineering, and economics.

Finding the Numbers: Plugging in the Values

Alright, we've done the heavy lifting, guys! We found two possible values for 'x': -5/3 and 5. Remember, 'x' represents our smaller number. So, let’s find the corresponding larger number (x + 5) for each of these values. If x = -5/3, then x + 5 = -5/3 + 5 = -5/3 + 15/3 = 10/3. So one pair of numbers is -5/3 and 10/3. If x = 5, then x + 5 = 5 + 5 = 10. So another pair of numbers is 5 and 10. We have now solved for all the numbers requested by the problem. We found both pairs of numbers, and it's time for the final check. Let's make sure these numbers fit the original problem. Remember, the sum of the squares of the numbers must be five times the square of the smaller number. For the pair -5/3 and 10/3: (-5/3)² + (10/3)² = 25/9 + 100/9 = 125/9. And 5 * (-5/3)² = 5 * 25/9 = 125/9. It checks out! For the pair 5 and 10: 5² + 10² = 25 + 100 = 125. And 5 * 5² = 5 * 25 = 125. This also checks out! Both pairs of numbers satisfy the problem's conditions. Isn't that cool? We have not only solved the math problem but also verified that our answers are correct. The satisfaction of verifying our answers confirms that we understood the problem. We understood the steps, the variables, and the relationships described in the original problem.

Conclusion: Wrapping Up and Next Steps

Congratulations, everyone! We've successfully solved the problem. The numbers are either 5 and 10, or -5/3 and 10/3. This was a great example of how we can translate word problems into algebraic equations and solve them step by step. We used our knowledge of algebra to build the equations, and we utilized factoring skills to arrive at the final answer. We've shown how variables, equations, and mathematical operations can be used to describe real-world relationships. This approach is not only useful for solving math problems but also develops your critical thinking and problem-solving skills, which is vital in many other fields. Keep practicing, and you'll become more confident in tackling these types of problems. Remember, the key is to understand the problem, define your variables, and follow the steps systematically. You can always come back to review this, or you can find other similar problems. Also, consider creating your own problems and then solving them. This can be a very effective way to reinforce all of the concepts.

So, what's next? Try similar problems on your own. Change the numbers, change the relationships, and see if you can still solve them. Look for patterns and practice different methods to find the solutions. There are many online resources and textbooks available to help you. And remember, the more you practice, the better you'll get. Consider checking other algebra topics like linear equations, inequalities, and functions. All these concepts are connected and will help you improve your overall math skills. Moreover, exploring other math concepts is a great way to improve your skills. Math is just a tool to understand the world, and it will open doors to new possibilities. Thanks for joining me on this math adventure. Keep learning, keep exploring, and keep practicing! Until next time, keep crunching those numbers!