Solving Equations (n^2 + 7n + 6) / N^2 = 1/6 - 1/(6n^2) Step-by-Step
Hey there, math enthusiasts! Ever stumbled upon an equation that looks like a tangled mess of variables and fractions? Well, you're not alone! Today, we're diving deep into solving one such equation, breaking it down step by step so that even the most complex-looking problems become a piece of cake. We will focus on solving this equation: (n^2 + 7n + 6) / n^2 = 1/6 - 1/(6n^2). So, grab your thinking caps, and let's get started!
The Challenge: A Tricky Equation
Before we jump into the solution, let's take a good look at what we're dealing with. Our equation is: (n^2 + 7n + 6) / n^2 = 1/6 - 1/(6n^2). At first glance, it might seem a bit daunting. We've got fractions, variables squared, and a mix of terms that could easily make anyone's head spin. But don't worry, we're going to tackle this methodically, turning this mathematical puzzle into a clear, understandable solution. Remember, the key to solving complex equations is to break them down into smaller, manageable parts. This approach not only makes the process less intimidating but also helps in identifying the best strategies to use.
Now, let's think about our plan of attack. What's the first thing that comes to mind when you see fractions in an equation? For me, it's getting rid of them! Clearing fractions often simplifies the equation and makes it easier to work with. So, that's going to be our primary goal. We'll also need to keep an eye out for any restrictions on our variable, n, since we have n^2 in the denominator. We can't let the denominator be zero, or we'll end up with an undefined expression. So, buckle up, guys, we're about to embark on a mathematical adventure!
Step 1: Clearing the Fractions
Our initial goal is to eliminate the fractions from the equation. To do this, we need to find the least common denominator (LCD) of all the fractions involved. Looking at our equation, (n^2 + 7n + 6) / n^2 = 1/6 - 1/(6n^2), we can identify the denominators as n^2, 6, and 6n^2. The LCD is the smallest expression that all these denominators can divide into evenly. In this case, the LCD is 6n^2. Now, we're going to multiply both sides of the equation by this LCD. This step is crucial because it will effectively clear the fractions, transforming our equation into a more manageable form. Remember, whatever we do to one side of the equation, we must do to the other to maintain balance. So, let's multiply both sides by 6n^2:
6n^2 * [(n^2 + 7n + 6) / n^2] = 6n^2 * [1/6 - 1/(6n^2)]
Now, we distribute the 6n^2 on both sides. On the left side, the n^2 in the numerator and denominator will cancel out, leaving us with 6 * (n^2 + 7n + 6). On the right side, we need to distribute 6n^2 to both terms inside the brackets. When we multiply 6n^2 by 1/6, the 6s cancel out, leaving us with n^2. When we multiply 6n^2 by -1/(6n^2), the 6n^2 terms completely cancel out, leaving us with -1. So, after distributing, our equation looks like this:
6(n^2 + 7n + 6) = n^2 - 1
See how much simpler the equation looks now? No more fractions! This is the power of clearing fractions – it transforms a complex-looking equation into something much more approachable. Our next step will be to further simplify this equation by distributing and combining like terms. We're on our way to solving this equation, one step at a time!
Step 2: Simplifying and Rearranging
Now that we've cleared the fractions, our equation looks much cleaner: 6(n^2 + 7n + 6) = n^2 - 1. The next step is to simplify both sides of the equation as much as possible. This typically involves distributing any coefficients and combining like terms. Let's start by distributing the 6 on the left side of the equation. We multiply 6 by each term inside the parentheses: 6 * n^2, 6 * 7n, and 6 * 6. This gives us 6n^2 + 42n + 36. So, our equation now looks like this:
6n^2 + 42n + 36 = n^2 - 1
Now that we've distributed, it's time to rearrange the equation. Our goal is to get all the terms on one side and set the equation equal to zero. This is a standard approach when dealing with quadratic equations (equations where the highest power of the variable is 2). To do this, we'll subtract n^2 from both sides and add 1 to both sides. This will move all the terms to the left side, leaving zero on the right. Subtracting n^2 from both sides gives us 5n^2 + 42n + 36 = -1. Then, adding 1 to both sides gives us:
5n^2 + 42n + 37 = 0
We now have a quadratic equation in standard form: ax^2 + bx + c = 0, where a = 5, b = 42, and c = 37. This is a crucial step because it allows us to use various methods to solve for n, such as factoring, completing the square, or using the quadratic formula. The key is to recognize the form of the equation and choose the most appropriate method for solving for the unknown variable. In this case, we'll explore factoring as our next step, but it's always good to have other methods in your toolkit in case factoring doesn't work out neatly. We are making progress towards understanding and solving equations like this!
Step 3: Factoring the Quadratic Equation
We've arrived at a quadratic equation: 5n^2 + 42n + 37 = 0. Now, let's see if we can solve it by factoring. Factoring involves breaking down the quadratic expression into two binomials (expressions with two terms) that, when multiplied together, give us the original quadratic. This can be a bit like solving a puzzle, but with practice, you'll get the hang of it. The general idea is to find two numbers that multiply to give the product of the leading coefficient (5) and the constant term (37), which is 5 * 37 = 185, and that add up to the middle coefficient (42). Finding these numbers might take a bit of trial and error, but it's a worthwhile effort if it leads to a solution.
After some thought, we can see that the numbers 5 and 37 fit the bill! 5 * 37 = 185 and 5 + 37 = 42. Now, we can rewrite the middle term (42n) using these two numbers. This process is called factoring by grouping. So, we rewrite our equation as:
5n^2 + 5n + 37n + 37 = 0
Notice how we've split the 42n term into 5n and 37n. Now, we group the first two terms and the last two terms together:
(5n^2 + 5n) + (37n + 37) = 0
Next, we factor out the greatest common factor (GCF) from each group. From the first group, we can factor out 5n, and from the second group, we can factor out 37. This gives us:
5n(n + 1) + 37(n + 1) = 0
Now, we see that (n + 1) is a common factor in both terms. We can factor it out:
(n + 1)(5n + 37) = 0
We've successfully factored the quadratic equation! This is a major step because it transforms our equation into a product of two factors equal to zero. According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. This gives us two separate equations to solve:
n + 1 = 0 or 5n + 37 = 0
We're almost there! In the next step, we'll solve these two simple equations to find the values of n that satisfy our original equation. Factoring is a powerful tool for solving quadratic equations, and mastering it will greatly enhance your mathematical abilities.
Step 4: Solving for n
We've factored our quadratic equation into (n + 1)(5n + 37) = 0. Now, we need to solve for n. As we discussed earlier, the zero-product property tells us that if the product of two factors is zero, then at least one of the factors must be zero. This gives us two separate equations to solve:
- n + 1 = 0
- 5n + 37 = 0
Let's solve the first equation, n + 1 = 0. To isolate n, we simply subtract 1 from both sides of the equation:
n = -1
So, one solution is n = -1. Now, let's move on to the second equation, 5n + 37 = 0. This one requires a little more work. First, we subtract 37 from both sides:
5n = -37
Next, we divide both sides by 5 to isolate n:
n = -37/5
So, our second solution is n = -37/5. We've found two potential solutions for n: -1 and -37/5. But before we declare victory, there's one crucial step we need to take: checking our solutions. It's always important to plug our solutions back into the original equation to make sure they work and don't lead to any undefined expressions (like division by zero). This is especially important when dealing with equations that have variables in the denominator, as we do in our original equation. Checking our solutions will ensure that we have valid answers and haven't made any mistakes along the way. We are getting closer to fully solving for n and understanding the solutions to this equation!
Step 5: Checking for Extraneous Solutions
We've found two potential solutions for our equation: n = -1 and n = -37/5. Now comes the crucial step of checking these solutions to make sure they are valid and don't lead to any undefined expressions in our original equation: (n^2 + 7n + 6) / n^2 = 1/6 - 1/(6n^2). Remember, whenever we have variables in the denominator, we need to be extra careful about extraneous solutions – solutions that we find algebraically but don't actually work when plugged back into the original equation. The main reason for this is that we cannot divide by zero. So, any value of n that makes the denominator zero is an extraneous solution.
Let's start by checking n = -1. We substitute -1 for n in our original equation:
((-1)^2 + 7(-1) + 6) / (-1)^2 = 1/6 - 1/(6(-1)^2)
Simplifying the left side, we get (1 - 7 + 6) / 1 = 0 / 1 = 0. Simplifying the right side, we get 1/6 - 1/(6 * 1) = 1/6 - 1/6 = 0. Since both sides are equal, n = -1 is a valid solution.
Now, let's check n = -37/5. This one is going to involve some fractions, but we'll take it step by step. Substituting -37/5 for n in our original equation, we get:
(((-37/5)^2 + 7(-37/5) + 6) / (-37/5)^2 = 1/6 - 1/(6(-37/5)^2)
This looks a bit intimidating, but let's simplify each side carefully. After a bit of arithmetic (which I won't show in every single detail here, but you can work it out on your own), you'll find that both sides of the equation are equal. This confirms that n = -37/5 is also a valid solution.
Since both potential solutions check out, we can confidently say that our solutions are n = -1 and n = -37/5. This process of checking for extraneous solutions is a vital part of equation solving, ensuring that we arrive at the correct answers. We have now successfully solved the equation!
Conclusion: The Thrill of the Solve
And there you have it, guys! We've successfully navigated a complex equation, breaking it down step by step and arriving at our solutions: n = -1 and n = -37/5. We started with a seemingly daunting equation, but by systematically clearing fractions, simplifying, factoring, and checking our solutions, we were able to conquer it. This journey highlights the power of methodical problem-solving in mathematics. Understanding equations and solving for unknowns is a fundamental skill that opens doors to more advanced mathematical concepts and real-world applications.
Remember, the key to tackling complex problems is to break them down into smaller, more manageable steps. Don't be intimidated by the initial appearance of an equation; instead, focus on identifying the underlying structure and choosing the appropriate techniques to apply. Whether it's clearing fractions, distributing, factoring, or using the quadratic formula, each step brings you closer to the solution. And most importantly, always remember to check your solutions to ensure they are valid!
So, the next time you encounter a tricky equation, remember the strategies we've discussed here. Embrace the challenge, break it down, and enjoy the thrill of the solve! Math is not just about finding the right answer; it's about the journey of discovery and the satisfaction of unraveling a puzzle. Keep practicing, keep exploring, and you'll become a master equation solver in no time! Solving equations is an essential part of mathematical learning and a valuable skill for life.