Solving X^2 + X - 12 = 0 Finding The Value Of X

Hey guys! Let's dive into a classic math problem today that involves solving a quadratic equation. Quadratic equations might sound intimidating, but they're actually quite manageable once you understand the basic principles. We're going to break down a specific problem step-by-step, making sure you grasp the concepts along the way. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into the solution, let's clearly state the problem we're tackling. We are given the quadratic equation x² + x - 12 = 0, and we know that x is a positive number. Our mission is to find the exact value of x that satisfies this equation. In simpler terms, we need to find the positive number that, when plugged into the equation, makes the equation true. Understanding the question is the first and most crucial step in solving any math problem. Without a clear grasp of what's being asked, we risk going down the wrong path. In this case, we know we're dealing with a quadratic equation, which means we're looking for values of x that make the equation equal to zero. The fact that x is positive is a crucial piece of information that will help us narrow down our solution later on. Quadratic equations are equations of the form ax² + bx + c = 0, where a, b, and c are constants. The solutions to these equations are also known as roots or zeros. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. For this particular problem, we'll focus on the factoring method, as it's often the quickest and most straightforward approach when applicable. Recognizing the structure of the equation and identifying the given conditions are essential skills in problem-solving. They allow us to choose the most efficient method and avoid unnecessary complications. Now that we have a firm understanding of the problem, let's move on to the exciting part: finding the solution!

Methods to Solve Quadratic Equations

Now, before we solve our specific equation, let's chat about the different ways we can tackle quadratic equations in general. It's like having a toolbox with different tools – knowing which one to use makes the job much easier! There are primarily three methods we can use: factoring, completing the square, and the quadratic formula. Each method has its strengths and is more suitable for certain types of quadratic equations. Let's briefly discuss each one. Factoring is often the first method we try because it's usually the quickest and simplest. Factoring involves breaking down the quadratic expression into two binomial expressions. For example, if we can rewrite x² + x - 12 as (x + a)(x + b), then we can easily find the values of x that make the equation zero. However, factoring isn't always possible, especially if the roots are not integers. Completing the square is a more versatile method that can be used to solve any quadratic equation. It involves manipulating the equation algebraically to create a perfect square trinomial. While it's a reliable method, it can be a bit more involved than factoring. Understanding how to complete the square is also crucial for deriving the quadratic formula. The quadratic formula is the most general method for solving quadratic equations. It can be used to solve any quadratic equation, regardless of whether it can be factored or not. The formula is given by x = [-b ± √(b² - 4ac)] / 2a, where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0. The quadratic formula is a powerful tool, but it's essential to remember it correctly and apply it carefully. The choice of method often depends on the specific equation we're dealing with. If the equation is easily factorable, factoring is the way to go. If not, completing the square or using the quadratic formula are reliable alternatives. Understanding the strengths and weaknesses of each method is crucial for efficient problem-solving. In our case, we'll see that factoring is the most straightforward approach, but it's good to have the other methods in our toolkit for future problems.

Solving by Factoring

Okay, let's get our hands dirty and solve the equation x² + x - 12 = 0 using factoring. Factoring, in this context, means we want to rewrite the quadratic expression as a product of two binomials. Think of it like reversing the process of expanding brackets. We are essentially looking for two numbers that, when multiplied, give us -12 (the constant term) and, when added, give us 1 (the coefficient of the x term). This is a classic puzzle-like approach that many people find quite satisfying. Let's break down the steps. We need to find two numbers that multiply to -12. Let's list the pairs of factors of 12: 1 and 12, 2 and 6, 3 and 4. Since we need a product of -12, one of the numbers in each pair must be negative. Now, we need to consider which of these pairs also add up to 1. This is where we start playing with the signs. If we try 3 and 4, we see that 4 - 3 = 1. Bingo! We've found our numbers. So, we can rewrite the equation as (x + 4)(x - 3) = 0. Notice how the +4 and -3 correspond to the numbers we found. This is the crucial step in factoring: correctly identifying the two numbers that satisfy both conditions. Now that we have the equation in factored form, it's much easier to find the solutions. The equation (x + 4)(x - 3) = 0 is true if either (x + 4) = 0 or (x - 3) = 0. This is because anything multiplied by zero is zero. Solving these two simple equations gives us x = -4 or x = 3. We now have two potential solutions for x. Factoring is a powerful technique because it transforms a quadratic equation into a simpler form that's easy to solve. The key is to practice identifying the correct factors and understanding the relationship between the factors and the coefficients of the quadratic equation. With practice, factoring becomes a quick and efficient method for solving many quadratic equations. Let's move on to the final step of choosing the correct solution.

Choosing the Correct Solution

Alright, we've done the heavy lifting and found two possible solutions for x: x = -4 and x = 3. But hold on! Remember a crucial piece of information from the original problem? It stated that x is a positive number. This is a common trick in math problems – they give you extra information that helps you narrow down the answers. This is where reading the question carefully truly pays off. We have two potential solutions, but only one of them fits the condition that x must be positive. x = -4 is a negative number, so we can eliminate it. This leaves us with x = 3 as the only valid solution. It's essential to always go back to the original problem and check if your solutions make sense in the context of the question. Sometimes, mathematical operations can lead to extraneous solutions – solutions that satisfy the equation but don't fit the original conditions. In this case, the condition that x is positive helps us filter out the extraneous solution. So, after carefully considering all the information, we can confidently say that the value of x that satisfies the equation x² + x - 12 = 0 and the condition that x is positive is x = 3. This final step of checking and choosing the correct solution is just as important as the algebraic manipulation. It ensures that we arrive at the correct answer and avoid making careless mistakes. Congratulations, guys! We've successfully solved the quadratic equation and found the value of x. Let's recap what we've done to solidify our understanding.

Conclusion: The Value of x

So, to wrap things up, we successfully navigated the world of quadratic equations and found the value of x in the equation x² + x - 12 = 0. We started by understanding the problem, exploring different methods for solving quadratic equations, and then settling on the factoring method. We skillfully factored the equation into (x + 4)(x - 3) = 0, which gave us two potential solutions: x = -4 and x = 3. But, and this is a big but, we remembered the crucial detail that x had to be a positive number. This allowed us to eliminate x = -4 and confidently declare that the solution is x = 3. This problem beautifully illustrates the importance of understanding the fundamental concepts of algebra, like factoring, and the critical role of careful reading and attention to detail. It's not just about manipulating equations; it's about understanding what the question is asking and using all the information provided to arrive at the correct answer. Remember, guys, practice makes perfect! The more you work with quadratic equations and other math problems, the more comfortable and confident you'll become. Don't be afraid to make mistakes – they're part of the learning process. Just keep practicing, keep asking questions, and keep exploring the fascinating world of mathematics. And who knows, maybe you'll even start to enjoy solving these puzzles as much as I do! Keep up the great work, and I'll see you in the next math adventure!