Solving 8x² - 24 = 3x² - 9 A Step-by-Step Guide
Introduction: Understanding Quadratic Equations
Quadratic equations are a fundamental part of algebra, and mastering them is crucial for success in higher mathematics. These equations, characterized by the presence of a squared term (x²), often appear in various fields like physics, engineering, and computer science. In this article, we will delve into the process of solving a specific quadratic equation: 8x² - 24 = 3x² - 9. We will break down each step, ensuring a clear understanding of the underlying concepts and techniques involved. By the end of this guide, you will not only be able to solve this particular equation but also gain a solid foundation for tackling other quadratic equations.
Before we jump into the solution, let's quickly recap what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (usually 'x') is 2. The general form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to 0. The equation we're about to solve, 8x² - 24 = 3x² - 9, fits this description, albeit in a slightly rearranged form. Our goal is to manipulate this equation to fit the standard form and then find the values of 'x' that satisfy the equation, which are also known as the roots or solutions of the equation. Understanding the nature of quadratic equations and the methods to solve them is a key skill in mathematics, opening doors to more advanced topics and real-world applications.
Step 1: Rearranging the Equation
In this initial step, our primary objective is to rearrange the given equation, 8x² - 24 = 3x² - 9, into a more standard and manageable form. The standard form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants. To achieve this, we need to consolidate all terms on one side of the equation, leaving zero on the other side. This process involves a series of algebraic manipulations, ensuring that we maintain the equation's balance and integrity.
The first action we take is to subtract 3x² from both sides of the equation. This step is crucial because it begins the process of grouping like terms together. By subtracting 3x² from both sides, we eliminate the x² term on the right side and bring it to the left side, which is a key step in achieving the standard form. The equation now looks like this: 8x² - 3x² - 24 = -9. This subtraction is a fundamental algebraic operation that helps simplify the equation and move us closer to isolating the variable terms. Next, we simplify the left side by combining the x² terms. 8x² minus 3x² equals 5x², which further simplifies the equation to 5x² - 24 = -9. This simplification is a crucial step as it reduces the number of terms and makes the equation easier to work with. Finally, we need to move the constant term from the right side to the left side. To do this, we add 9 to both sides of the equation. This action isolates all terms on the left side and leaves zero on the right side, which is the hallmark of the standard form of a quadratic equation. Adding 9 to both sides gives us 5x² - 24 + 9 = 0. Simplifying this, we get our rearranged equation: 5x² - 15 = 0. This rearranged equation is now in a much more convenient form for solving, closely resembling the standard quadratic form. This step of rearranging the equation is not just about mathematical manipulation; it's about setting up the problem in a way that makes the solution process clearer and more straightforward. By understanding this step, we lay a solid foundation for the subsequent steps in solving the quadratic equation.
Step 2: Isolating the x² Term
After successfully rearranging our equation to the form 5x² - 15 = 0, the next crucial step is to isolate the x² term. Isolating the x² term means getting it by itself on one side of the equation. This is a pivotal step because once we have the x² term isolated, we can then easily solve for x by taking the square root. The process involves performing algebraic operations that maintain the balance of the equation while gradually stripping away the terms surrounding the x².
Our starting point is the equation 5x² - 15 = 0. The first action we take to isolate the x² term is to add 15 to both sides of the equation. This is done to eliminate the constant term (-15) from the left side, bringing us closer to having the x² term alone. Adding 15 to both sides gives us 5x² = 15. This step is a fundamental algebraic manipulation that preserves the equality of the equation while simplifying it. Now that we have 5x² = 15, the next step is to get rid of the coefficient (5) that is multiplying the x² term. To do this, we divide both sides of the equation by 5. This operation is the key to isolating the x² term completely. Dividing both sides by 5, we get x² = 3. At this point, we have successfully isolated the x² term. The equation is now in a simple form where x² is equal to a constant. This isolation is a significant achievement in the solving process because it sets the stage for the final step: finding the values of x that satisfy this equation. Understanding the logic behind isolating the x² term is crucial. It's not just about following steps; it's about strategically manipulating the equation to bring the variable term into focus. This step demonstrates the power of algebraic operations in simplifying complex equations and making them solvable.
Step 3: Solving for x
Having isolated the x² term, our final and most exciting step is to solve for x in the equation x² = 3. This step involves undoing the square operation to find the values of x that satisfy the equation. The fundamental principle we employ here is the inverse operation of squaring, which is taking the square root. However, a critical aspect to remember when taking the square root in the context of solving equations is that there are typically two possible solutions: a positive root and a negative root. This arises because both the positive and negative values, when squared, will yield the same positive result.
To solve for x in x² = 3, we take the square root of both sides of the equation. This operation is the key to unraveling the square on x and revealing the possible values of x. When we take the square root of x², we get x. However, when we take the square root of 3, we must consider both the positive and negative roots, as both (√3)² and (-√3)² equal 3. Therefore, we write x = ±√3. This notation is a concise way of representing the two possible solutions: x = √3 and x = -√3. These are the two values that, when squared, will give us 3, thus satisfying the original equation. Understanding why we consider both positive and negative roots is crucial for a complete understanding of solving quadratic equations. It's not just about finding one value that works; it's about recognizing the dual nature of squaring and its inverse operation. The solutions x = √3 and x = -√3 are the exact solutions to the equation. If we need to express these solutions as decimal approximations, we can use a calculator to find that √3 is approximately 1.732. Therefore, the approximate solutions are x ≈ 1.732 and x ≈ -1.732. These decimal approximations provide a more tangible sense of the values of x that satisfy the equation. In summary, solving for x in x² = 3 involves taking the square root of both sides, remembering to account for both positive and negative roots. This step is a culmination of all the previous steps, bringing us to the final solution of the quadratic equation. The solutions x = √3 and x = -√3 are the values that make the equation 8x² - 24 = 3x² - 9 true, and understanding how we arrived at these solutions is key to mastering quadratic equations.
Conclusion: Summarizing the Solution
In this comprehensive guide, we have successfully navigated the process of solving the quadratic equation 8x² - 24 = 3x² - 9. We began by understanding the fundamentals of quadratic equations, recognizing their standard form, and appreciating their significance in various fields of study. Then, we methodically broke down the solution into three key steps, each building upon the previous one to lead us to the final answer. Let's take a moment to recap these steps and highlight the key concepts involved.
Step 1 was all about rearranging the equation. We transformed the original equation into the standard-like form, 5x² - 15 = 0. This involved strategically adding and subtracting terms on both sides of the equation to consolidate like terms and move all terms to one side, leaving zero on the other. This step is crucial because it sets up the equation in a format that is conducive to further manipulation and solving. Understanding how to rearrange equations is a fundamental skill in algebra, applicable to a wide range of problems beyond just quadratic equations.
Step 2 focused on isolating the x² term. Starting from 5x² - 15 = 0, we added 15 to both sides and then divided by 5 to obtain x² = 3. This isolation is a pivotal step because it allows us to directly address the variable we are trying to solve for. Isolating terms is a common strategy in algebra, and mastering this technique is essential for solving various types of equations. It's about strategically manipulating the equation to bring the variable into focus.
Step 3 was the final step, where we solved for x. Taking the square root of both sides of x² = 3, we arrived at the solutions x = √3 and x = -√3. It's crucial to remember that taking the square root introduces both positive and negative solutions because both (√3)² and (-√3)² equal 3. This understanding is vital for a complete solution of quadratic equations. The solutions x = √3 and x = -√3 are the values that satisfy the original equation. They are the points where the quadratic function represented by the equation intersects the x-axis. In decimal form, these solutions are approximately x ≈ 1.732 and x ≈ -1.732.
In conclusion, solving the quadratic equation 8x² - 24 = 3x² - 9 involved a systematic approach of rearranging, isolating, and then solving for the variable. Each step required a careful application of algebraic principles, and the final solution highlighted the importance of considering both positive and negative roots when taking square roots. This process not only provides the solutions to the specific equation but also reinforces the broader skills of algebraic manipulation and problem-solving, which are invaluable in mathematics and beyond. By understanding and practicing these techniques, you can confidently tackle a wide range of quadratic equations and related problems.
