Introduction
Hey guys! Today, we're diving into a fun little math problem that involves solving a quadratic equation. Quadratic equations might seem intimidating at first, but trust me, once you get the hang of them, they're not so bad. We're going to break down the equation 5x² + 16 = 7x²/3 step by step, so you can see exactly how to solve it. This type of problem is a classic example of what you might encounter in algebra, and mastering it will definitely boost your math skills. We’ll cover everything from simplifying the equation to finding the roots, so buckle up and let's get started!
In this article, we'll explore a particular quadratic equation: 5x² + 16 = 7x²/3. We will walk through the step-by-step process to solve it, ensuring that you understand each transformation and calculation. Quadratic equations are an essential topic in algebra, and they appear in various real-world applications, from physics to engineering. By mastering the techniques to solve these equations, you'll not only improve your math skills but also gain a valuable tool for problem-solving in other fields. The initial equation looks a bit complex with the fraction, but don't worry! Our first step will be to eliminate that fraction to make the equation easier to work with. We'll then rearrange the terms to get the standard form of a quadratic equation, which is ax² + bx + c = 0. Once we have the equation in this form, we can determine the best method to solve it. In this case, we'll see that factoring isn't straightforward, and we don't have a simple 'b' term to complete the square easily. Therefore, we'll use the square root property, which is perfectly suited for equations where we can isolate the squared term. By following these steps, we'll systematically find the values of x that satisfy the equation. So, let's dive in and start with the first step: eliminating the fraction!
Step 1: Eliminating the Fraction
Okay, so the first thing we need to do is get rid of that pesky fraction. Fractions in equations can make things look more complicated than they are, so let's simplify our lives by eliminating it. Our equation is 5x² + 16 = 7x²/3. To get rid of the denominator, which is 3 in this case, we're going to multiply both sides of the equation by 3. Remember, whatever you do to one side of the equation, you have to do to the other side to keep things balanced. So, we multiply both (5x² + 16) and (7x²/3) by 3. This gives us 3 * (5x² + 16) = 3 * (7x²/3). Now, we distribute the 3 on the left side: 3 * 5x² + 3 * 16, which simplifies to 15x² + 48. On the right side, the 3 in the numerator and the 3 in the denominator cancel each other out, leaving us with just 7x². So, our equation now looks much cleaner: 15x² + 48 = 7x². See? Much better already! Getting rid of the fraction makes the equation more manageable, and we're one step closer to solving it. Now that we've eliminated the fraction, the next step is to rearrange the equation so that all the x² terms are on one side and the constants are on the other. This will help us isolate x and eventually find its value. So, let’s move on to the next step and continue simplifying our equation. Remember, the key to solving these problems is to take it one step at a time, and you'll get there!
Eliminating fractions is a crucial initial step when solving equations, especially those involving rational expressions. In our case, multiplying both sides of the equation 5x² + 16 = 7x²/3 by 3 simplifies the equation significantly. By doing this, we transform the original equation into a more manageable form without altering its solutions. This technique is based on the fundamental principle that multiplying both sides of an equation by the same non-zero number preserves the equality. When we multiply the left side, (5x² + 16), by 3, we apply the distributive property to get 3 * 5x² + 3 * 16, which simplifies to 15x² + 48. On the right side, the multiplication by 3 cancels out the denominator, resulting in 7x². Thus, the equation becomes 15x² + 48 = 7x², a form that is much easier to work with. This step is essential because it removes the complexity introduced by the fraction, allowing us to proceed with rearranging and solving the equation more efficiently. By clearing the fraction, we pave the way for the next step, which involves grouping like terms and isolating the variable. This approach is a standard technique in algebra, and mastering it will help you tackle a variety of equations with greater confidence. So, let’s move forward and see how we can further simplify the equation to find the value of x.
Step 2: Rearranging the Equation
Alright, now that we've gotten rid of the fraction, the next step is to rearrange the equation so that all the x² terms are on one side and the constant terms are on the other. This will help us isolate x and eventually solve for its value. Our equation is currently 15x² + 48 = 7x². We want to get all the x² terms together, so let's subtract 7x² from both sides of the equation. This gives us 15x² - 7x² + 48 = 7x² - 7x². Simplifying this, we get 8x² + 48 = 0. Now, we want to isolate the x² term, so let's subtract 48 from both sides of the equation. This gives us 8x² + 48 - 48 = 0 - 48, which simplifies to 8x² = -48. We're getting closer! We now have the x² term almost by itself. The next step is to divide both sides by 8 to completely isolate x². So, let’s do that in the next step. Remember, rearranging equations is all about keeping things balanced and doing the same operation on both sides. By methodically moving terms around, we can simplify the equation and make it easier to solve. So, let's keep going and see what the next step brings!
Rearranging the equation is a critical step in solving for the variable. This process involves moving terms around while maintaining the equality of both sides. Starting with our equation, 15x² + 48 = 7x², the first goal is to group the x² terms together. To do this, we subtract 7x² from both sides. This gives us 15x² - 7x² + 48 = 7x² - 7x², which simplifies to 8x² + 48 = 0. This step is crucial because it consolidates the variable terms on one side, making the equation easier to manipulate. Next, we want to isolate the x² term further. To achieve this, we subtract the constant term, 48, from both sides of the equation. This results in 8x² + 48 - 48 = 0 - 48, which simplifies to 8x² = -48. Now, we have the x² term almost entirely isolated. The equation is much simpler than when we started, and we’re closer to finding the value of x. This step-by-step approach of rearranging terms is a fundamental technique in algebra. By carefully applying these steps, we can transform complex equations into simpler forms that are easier to solve. Each step brings us closer to isolating the variable and finding its value. So, let’s continue to the next step, where we will completely isolate the x² term and prepare to solve for x.
Step 3: Isolating x²
Okay, we're on the home stretch now! We've done a great job simplifying the equation so far. We're currently at 8x² = -48. Our goal here is to isolate x², which means we need to get rid of the 8 that's multiplying it. To do this, we'll divide both sides of the equation by 8. Remember, whatever we do to one side, we have to do to the other to keep things balanced. So, we have (8x²)/8 = -48/8. On the left side, the 8s cancel each other out, leaving us with just x². On the right side, -48 divided by 8 is -6. So, our equation now looks like this: x² = -6. We've successfully isolated x²! This is a huge step because now we can see what x² equals. But we're not quite done yet. We still need to find the value of x, not just x². To do that, we'll need to take the square root of both sides. But before we do that, let's pause for a moment and think about what x² = -6 means. Can a square of a real number be negative? This is a crucial question that will determine the nature of our solutions. So, let's keep going and see how we handle this in the next step!
Isolating x² is a pivotal step in solving for x. From our previous rearrangement, we have the equation 8x² = -48. To isolate x², we need to eliminate the coefficient 8. This is achieved by dividing both sides of the equation by 8. So, we perform the operation (8x²)/8 = -48/8. On the left side, the 8 in the numerator and the 8 in the denominator cancel each other out, leaving us with x². On the right side, -48 divided by 8 equals -6. Therefore, our equation simplifies to x² = -6. At this point, we have successfully isolated x². This simplification is significant because it sets the stage for the final step of solving for x. By isolating x², we can now easily see the relationship between x² and a numerical value. However, this result also raises an important consideration: we have x² equal to a negative number. This means that when we take the square root of both sides to solve for x, we will be dealing with the square root of a negative number, which leads us into the realm of complex numbers. So, the next step will involve taking the square root of both sides, and we will need to remember the properties of complex numbers to find the solutions for x. Let’s proceed to the next step and explore how we handle this situation.
Step 4: Taking the Square Root
Alright, we've reached a crucial point in our problem. We've got x² = -6. Now, to find x, we need to take the square root of both sides of the equation. Remember, when you take the square root of a variable squared, you get two possible solutions: a positive and a negative. This is because both the positive and negative values, when squared, will give you the same positive result. So, taking the square root of both sides, we get √(x²) = ±√(-6). On the left side, the square root of x² is simply x. On the right side, we have the square root of -6. Now, here's where things get a little interesting. We can't take the square root of a negative number in the real number system. This means our solutions will be complex numbers. Remember that the square root of -1 is defined as the imaginary unit, i. So, we can rewrite √(-6) as √(6 * -1), which is the same as √(6) * √(-1), and that equals √(6)i. Therefore, our equation becomes x = ±√(6)i. So, we have two solutions for x: x = √(6)i and x = -√(6)i. These are complex solutions because they involve the imaginary unit i. We've successfully found the values of x that satisfy our original equation! This problem demonstrates how quadratic equations can have complex solutions, and it's a great example of how math can take us into fascinating new territories. So, let's recap our steps and make sure we've got everything clear.
Taking the square root is the final step in solving for x when we have isolated x². Our equation is x² = -6. To find x, we need to take the square root of both sides. When we do this, we must remember that there are two possible solutions: a positive square root and a negative square root. This is because both the positive and negative values, when squared, will yield the same positive result. Thus, we write √(x²) = ±√(-6). The square root of x² is simply x. On the right side, we encounter the square root of a negative number, √(-6). In the real number system, we cannot take the square root of a negative number. However, we can express this in terms of complex numbers. Recall that the imaginary unit, i, is defined as √(-1). We can rewrite √(-6) as √(6 * -1), which can be further separated into √(6) * √(-1). Since √(-1) = i, we have √(-6) = √(6)i. Therefore, our equation becomes x = ±√(6)i. This means we have two complex solutions for x: x = √(6)i and x = -√(6)i. These solutions indicate that there are no real number solutions for the original equation, but there are two complex number solutions. This example illustrates the importance of understanding complex numbers when solving quadratic equations. We have successfully found the values of x that satisfy the equation, even though they are complex. Let’s now summarize the entire process to ensure we have a clear understanding of each step.
Conclusion
So, guys, we've made it to the end! We started with the equation 5x² + 16 = 7x²/3 and went through a series of steps to solve for x. Let's quickly recap what we did. First, we eliminated the fraction by multiplying both sides of the equation by 3, which gave us 15x² + 48 = 7x². Then, we rearranged the equation to get all the x² terms on one side and the constants on the other, resulting in 8x² = -48. Next, we isolated x² by dividing both sides by 8, which gave us x² = -6. Finally, we took the square root of both sides, remembering to consider both the positive and negative roots. Since we had the square root of a negative number, we expressed our solutions in terms of the imaginary unit i, giving us the complex solutions x = ±√(6)i. And that's it! We've successfully solved the quadratic equation. This problem is a great example of how to handle equations with fractions and how to deal with complex solutions. Remember, practice makes perfect, so keep working on these types of problems, and you'll become a pro in no time. Quadratic equations are a fundamental part of algebra, and understanding how to solve them will be incredibly helpful in your math journey. Keep up the great work, and I'll see you in the next math adventure!
Throughout this article, we’ve systematically solved the quadratic equation 5x² + 16 = 7x²/3. We began by eliminating the fraction, which simplified the equation and made it easier to work with. We then rearranged the terms to isolate the x² term. This involved moving all x² terms to one side and the constant terms to the other, which is a crucial step in solving many algebraic equations. Once we had 8x² = -48, we isolated x² by dividing both sides by 8, resulting in x² = -6. This step set the stage for the final solution, but it also introduced an interesting twist: the square of x is a negative number. This indicates that the solutions will be complex numbers. Taking the square root of both sides, we obtained x = ±√(-6). Recognizing that √(-6) can be expressed as √(6)i, where i is the imaginary unit, we found the solutions to be x = √(6)i and x = -√(6)i. These complex solutions highlight the importance of understanding the broader number system beyond just real numbers. This process not only provides the solutions to this specific equation but also demonstrates a methodical approach to solving quadratic equations in general. By following these steps, you can tackle a variety of algebraic problems with confidence. So, remember to practice these techniques and apply them to other equations. The more you practice, the more comfortable you will become with solving quadratic equations and the more you will appreciate the beauty and power of mathematics.
