Solving (2x + 1)² = 49 A Comprehensive Step-by-Step Guide
Introduction
Hey guys! Today, we're diving into an exciting math problem: solving the equation (2x + 1)² = 49. This type of equation, involving a squared term, might seem intimidating at first glance, but don't worry! We'll break it down into simple, manageable steps. Think of it as a puzzle – each step gets us closer to the solution. We'll explore different methods to tackle this problem, ensuring you not only get the correct answer but also understand why each step is necessary. Math isn't just about numbers; it's about understanding the logic and the process. This equation falls under the realm of algebra, specifically quadratic equations (though we'll solve it without directly using the quadratic formula). Mastering these techniques is super important as they form the foundation for more advanced math concepts you'll encounter later. So, grab your pencils, notebooks, and let's embark on this mathematical journey together! We're going to cover everything from the basic principles involved to the practical steps you need to take to arrive at the solution. Remember, the key to math is practice, so the more you engage with these problems, the more confident you'll become. This guide is designed to be your friendly companion, walking you through each stage of the solution process. Whether you're a student tackling homework, a math enthusiast looking to sharpen your skills, or just curious about how these equations work, this guide is for you.
Method 1: Taking the Square Root
Okay, let's get started with our first method: taking the square root. This is a classic and often the most straightforward way to solve equations where you have a squared term isolated on one side. The core idea here is to reverse the squaring operation. We know that the square root of a number, when multiplied by itself, gives us the original number. For example, the square root of 9 is 3 because 3 * 3 = 9. However, there's a crucial thing to remember: both positive and negative numbers, when squared, result in a positive number. So, the square root of 9 could also be -3 because (-3) * (-3) = 9. This is why when we take the square root in an equation, we need to consider both the positive and negative roots. In our equation, (2x + 1)² = 49, we have (2x + 1) squared. To undo this square, we take the square root of both sides of the equation. This gives us two possibilities: 2x + 1 equals the positive square root of 49, which is 7, or 2x + 1 equals the negative square root of 49, which is -7. This split into two possibilities is the key to solving this equation correctly. We now have two simpler equations to solve: 2x + 1 = 7 and 2x + 1 = -7. Each of these equations will give us a different solution for x. Solving these linear equations involves isolating x, which we'll do by first subtracting 1 from both sides and then dividing by 2. It's important to remember to perform the same operation on both sides of the equation to maintain the balance. This method highlights a fundamental principle in algebra: performing inverse operations to isolate the variable we're trying to solve for. By taking the square root, we've effectively undone the squaring operation, paving the way to find the values of x that satisfy the original equation. This approach is not only efficient but also provides a clear understanding of the underlying mathematical concepts.
Step 1: Take the square root of both sides
The very first thing we need to do, as we discussed, is to take the square root of both sides of our equation, (2x + 1)² = 49. Remember, this is the crucial step in reversing the squaring operation. When we do this, we get √[(2x + 1)²] = ±√[49]. Notice the ± (plus or minus) sign in front of the square root of 49. This is absolutely essential because, as we talked about, both positive and negative numbers, when squared, yield a positive result. The square root of 49 is 7, so we have two possibilities to consider: 2x + 1 = 7 and 2x + 1 = -7. By acknowledging both possibilities, we ensure we capture all possible solutions for x. It's like unlocking two different paths to the answer. Now, let's think about what the square root operation actually does. When we take the square root of (2x + 1)², we're essentially asking, “What number, when multiplied by itself, equals (2x + 1)²?” The answer, of course, is (2x + 1). So, the left side of our equation simplifies to 2x + 1. This simplification is a direct result of the inverse relationship between squaring and taking the square root. This step neatly transforms our original equation into two simpler, linear equations that we can easily solve. It’s like taking a complex puzzle and breaking it down into smaller, more manageable pieces. By correctly applying the square root to both sides and remembering the ± sign, we've set ourselves up for a successful solution. This step is a great illustration of how mathematical operations can be used to simplify problems and reveal the underlying structure.
Step 2: Solve for the positive root
Now that we've taken the square root and have our two possibilities, let's solve for the positive root first. This means we're focusing on the equation 2x + 1 = 7. Our goal here is to isolate x, which means getting it all by itself on one side of the equation. To do this, we'll use the basic principles of algebraic manipulation. Remember, whatever operation we perform on one side of the equation, we must perform on the other side to maintain the balance. Think of an equation like a scale – if you add or remove weight from one side, you need to do the same on the other to keep it level. The first step in isolating x in the equation 2x + 1 = 7 is to get rid of the +1. We can do this by subtracting 1 from both sides of the equation. This gives us: 2x + 1 - 1 = 7 - 1, which simplifies to 2x = 6. See how the +1 on the left side has disappeared? We're one step closer to getting x alone. Next, we need to get rid of the 2 that's multiplying x. The opposite of multiplication is division, so we'll divide both sides of the equation by 2. This gives us: (2x) / 2 = 6 / 2, which simplifies to x = 3. And there we have it! We've solved for x in the positive root case, and we've found that x = 3 is one solution to our original equation. It's a great feeling when you isolate the variable and discover its value. This process demonstrates the power of using inverse operations to unravel an equation and reveal the hidden solution. We've carefully and methodically manipulated the equation, ensuring each step is logically sound and leads us closer to the answer.
Step 3: Solve for the negative root
Alright, we've tackled the positive root, and now it's time to solve for the negative root. This means we're now working with the equation 2x + 1 = -7. Remember, considering both the positive and negative roots is crucial for finding all possible solutions to our original equation, (2x + 1)² = 49. Just like before, our mission is to isolate x, getting it by itself on one side of the equation. We'll follow the same principles of algebraic manipulation we used for the positive root, making sure to maintain the balance of the equation at every step. The first order of business is to get rid of the +1 on the left side. We do this by subtracting 1 from both sides of the equation: 2x + 1 - 1 = -7 - 1. This simplifies to 2x = -8. Notice that subtracting 1 from -7 results in -8. It's important to pay close attention to the signs when working with negative numbers. We're now one step closer to isolating x. Next, we need to deal with the 2 that's multiplying x. As before, we'll use the inverse operation of multiplication, which is division. We divide both sides of the equation by 2: (2x) / 2 = -8 / 2. This simplifies to x = -4. And there we have it! We've found another solution for x: x = -4. This is the solution that corresponds to the negative root of 49. By systematically working through the equation, we've successfully uncovered another piece of the puzzle. Solving for the negative root highlights the importance of considering all possibilities when dealing with square roots. It's easy to overlook the negative root, but doing so would mean missing a valid solution. This process reinforces the idea that mathematical problem-solving is about careful attention to detail and a methodical approach.
Method 2: Expanding and Factoring
Let's explore another method to solve the equation (2x + 1)² = 49: expanding and factoring. This method takes a slightly different approach compared to taking the square root directly. Instead of immediately reversing the square, we'll first expand the squared term, then rearrange the equation into a standard quadratic form, and finally factorize it to find the solutions. This method is a great way to reinforce your understanding of algebraic manipulation and factoring techniques. Expanding (2x + 1)² means multiplying it by itself: (2x + 1)(2x + 1). We can use the FOIL method (First, Outer, Inner, Last) or the distributive property to carry out this multiplication. This will result in a quadratic expression, which is an expression of the form ax² + bx + c. Once we have this quadratic expression, we'll subtract 49 from both sides of the equation to set it equal to zero. This is a crucial step because factoring is typically done on quadratic equations in the standard form ax² + bx + c = 0. Factoring involves breaking down the quadratic expression into two binomial factors. This is like reversing the expansion process. The goal is to find two expressions that, when multiplied together, give us the original quadratic expression. Once we have the factored form, we can use the zero-product property to find the solutions for x. The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. This means we can set each factor equal to zero and solve for x. This method provides a solid understanding of the relationship between the expanded form of a quadratic equation and its factored form. It also showcases the power of factoring as a technique for solving equations. While it might seem a bit more involved than taking the square root directly in this particular case, it's a valuable skill to have in your mathematical toolkit, especially for more complex quadratic equations.
Step 1: Expand the left side of the equation
The first step in this method is to expand the left side of the equation, which is (2x + 1)². Remember, expanding means multiplying the expression by itself: (2x + 1)(2x + 1). We can use a couple of different techniques to do this, but the most common ones are the FOIL method and the distributive property. Let's use the FOIL method first. FOIL stands for First, Outer, Inner, Last, and it's a handy way to remember how to multiply two binomials. * First: Multiply the first terms in each binomial: (2x)(2x) = 4x². * Outer: Multiply the outer terms in the binomials: (2x)(1) = 2x. * Inner: Multiply the inner terms in the binomials: (1)(2x) = 2x. * Last: Multiply the last terms in each binomial: (1)(1) = 1. Now, we add these terms together: 4x² + 2x + 2x + 1. We can simplify this by combining the like terms (the 2x terms): 4x² + 4x + 1. So, (2x + 1)² expands to 4x² + 4x + 1. Alternatively, we could use the distributive property. This involves distributing each term in the first binomial to each term in the second binomial. So, we would multiply 2x by (2x + 1) and then multiply 1 by (2x + 1): 2x(2x + 1) + 1(2x + 1) = 4x² + 2x + 2x + 1. Again, simplifying by combining like terms gives us 4x² + 4x + 1. Either way, we arrive at the same result. Expanding the left side of the equation transforms it from a squared expression into a quadratic expression in the standard form. This is a crucial step in preparing the equation for factoring. It allows us to see the quadratic structure more clearly and apply the appropriate factoring techniques. This process highlights the importance of understanding different algebraic techniques and being able to choose the one that best suits the problem.
Step 2: Rewrite the equation in standard quadratic form
Now that we've expanded the left side of the equation, we need to rewrite the equation in standard quadratic form. This form is ax² + bx + c = 0, where a, b, and c are constants. Having the equation in this form is essential for factoring and applying other quadratic equation-solving techniques. Our expanded equation is 4x² + 4x + 1 = 49. To get it into the standard quadratic form, we need to move the 49 from the right side to the left side, so that the right side becomes zero. We can do this by subtracting 49 from both sides of the equation. Remember, whatever we do to one side, we must do to the other to maintain the balance. Subtracting 49 from both sides gives us: 4x² + 4x + 1 - 49 = 49 - 49. This simplifies to 4x² + 4x - 48 = 0. Now, our equation is in the standard quadratic form! We have a quadratic expression (4x² + 4x - 48) set equal to zero. Before we proceed to factoring, it's often a good idea to see if we can simplify the equation further. In this case, we notice that all the coefficients (4, 4, and -48) are divisible by 4. Dividing both sides of the equation by 4 will simplify the expression without changing the solutions. Dividing by 4 gives us: (4x² + 4x - 48) / 4 = 0 / 4, which simplifies to x² + x - 12 = 0. This simplified quadratic equation is much easier to factor. Rewriting the equation in standard quadratic form is a key step in solving quadratic equations. It allows us to apply the tools and techniques specifically designed for this form, such as factoring, the quadratic formula, or completing the square. This process demonstrates the importance of algebraic manipulation in transforming equations into a more manageable form.
Step 3: Factor the quadratic equation
Alright, we've got our equation in standard quadratic form: x² + x - 12 = 0. Now it's time to factor the quadratic equation. Factoring involves breaking down the quadratic expression into two binomials that, when multiplied together, give us the original expression. Think of it like reverse-engineering the expansion process we did earlier. To factor x² + x - 12, we need to find two numbers that: * Multiply to give the constant term (-12). * Add up to give the coefficient of the x term (which is 1 in this case). Let's think about the factors of -12. We have: * 1 and -12 * -1 and 12 * 2 and -6 * -2 and 6 * 3 and -4 * -3 and 4 Out of these pairs, the pair that adds up to 1 is -3 and 4. So, we can factor the quadratic expression as (x - 3)(x + 4). If you're unsure, you can always multiply these binomials back together using the FOIL method to check if you get the original expression. Now we have our factored equation: (x - 3)(x + 4) = 0. This is a significant step because it allows us to use the zero-product property, which we'll discuss in the next step. Factoring is a crucial skill in algebra, and it's a powerful technique for solving quadratic equations. It's like unlocking a secret code that reveals the solutions hidden within the equation. Mastering factoring requires practice and a good understanding of number relationships. By finding the right factors, we've transformed a complex equation into a simpler form that we can easily solve.
Step 4: Apply the zero-product property and solve for x
We've successfully factored our quadratic equation, and now we're at the final stage: applying the zero-product property and solving for x. The zero-product property is a fundamental principle in algebra that states: if the product of two factors is zero, then at least one of the factors must be zero. In our case, we have the equation (x - 3)(x + 4) = 0. This means that either (x - 3) = 0 or (x + 4) = 0 (or both). This property allows us to break our single equation into two simpler equations, each of which we can easily solve for x. Let's start with the first factor: x - 3 = 0. To solve for x, we simply add 3 to both sides of the equation: x - 3 + 3 = 0 + 3, which gives us x = 3. So, one solution is x = 3. Now, let's move on to the second factor: x + 4 = 0. To solve for x, we subtract 4 from both sides of the equation: x + 4 - 4 = 0 - 4, which gives us x = -4. So, our second solution is x = -4. And there we have it! We've found both solutions to our original equation using the expanding and factoring method. We arrived at the same solutions we found using the square root method, which is a good confirmation that our work is correct. Applying the zero-product property is the key step in unlocking the solutions once we've factored the quadratic equation. It's a direct and elegant way to find the values of x that make the equation true. This process showcases the power of factoring and the zero-product property as a combined technique for solving quadratic equations.
Conclusion
So, guys, we've successfully navigated the equation (2x + 1)² = 49 using two different methods: taking the square root and expanding and factoring. Both methods led us to the same solutions: x = 3 and x = -4. This highlights a really cool aspect of math – often, there's more than one way to arrive at the correct answer. The method you choose might depend on the specific problem, your personal preference, or what you find easiest to understand. Taking the square root is often the quicker method when you have a squared term isolated, as it directly undoes the squaring operation. However, expanding and factoring is a valuable skill to have in your toolkit, especially for more complex quadratic equations that might not be easily solved by taking the square root. Understanding both methods not only helps you solve this particular type of equation but also strengthens your overall algebraic skills. It's like having different tools in a toolbox – the more tools you have, the better equipped you are to tackle any job. Remember, the key to mastering math is practice. The more you work through different types of problems, the more comfortable and confident you'll become. Don't be afraid to try different approaches and see what works best for you. Math is a journey of discovery, and every problem you solve is a step forward. So, keep practicing, keep exploring, and most importantly, keep having fun with math! We've covered some fundamental concepts and techniques in this guide, and these will serve you well as you continue your mathematical adventures. Whether you're solving equations, graphing functions, or tackling more advanced topics, the skills you've learned here will provide a solid foundation for success.
