Finding The Value Of X In X² + 12x + 36 = 0 A Detailed Solution

Hey guys! Today, let's dive into a classic math problem: solving a quadratic equation. Quadratic equations might seem intimidating at first, but with a few tricks and some practice, you'll be solving them like a pro. We're going to tackle the equation x² + 12x + 36 = 0. Our mission is to find the value (or values) of x that make this equation true. We'll explore different methods, break down each step, and make sure you understand the why behind the how. So, grab your pencils and let's get started!

Understanding Quadratic Equations

Before we jump into solving, let's make sure we're all on the same page about what a quadratic equation actually is. A quadratic equation is a polynomial equation of the second degree. That might sound like a mouthful, but it just means the highest power of the variable (x in our case) is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants (numbers). Think of a as the coefficient of the x² term, b as the coefficient of the x term, and c as the constant term. These constants play a crucial role in determining the solutions to the equation. Understanding the structure of a quadratic equation is the first step to solving it. It helps us identify the different parts and choose the right method to find the solutions. In our equation, x² + 12x + 36 = 0, we can see that a is 1 (since there's no number explicitly written before x², it's understood to be 1), b is 12, and c is 36. Keep these values in mind as we move forward, as they'll be essential for the methods we'll use. Recognizing the coefficients is like having the key ingredients before starting a recipe – you need them to create the final product. This foundation will allow us to explore various solution methods with confidence and clarity. So, with this understanding, we're well-prepared to tackle the challenge of finding the value of x.

Methods to Solve Quadratic Equations

Now that we've got the basics down, let's talk about the different ways we can solve quadratic equations. There are a few main methods, each with its own strengths and weaknesses. The most common methods are factoring, using the quadratic formula, and completing the square. Factoring is like finding the building blocks of the equation. We try to rewrite the quadratic expression as a product of two binomials (expressions with two terms). This method is quick and easy when it works, but it's not always applicable. Some quadratic equations just don't factor nicely. The quadratic formula is a more general method. It's a formula that gives you the solutions directly, regardless of whether the equation factors or not. It might look a bit intimidating at first, but it's a reliable workhorse. Completing the square is another method that involves manipulating the equation to create a perfect square trinomial. It's less commonly used for solving equations directly, but it's an important technique for understanding the quadratic formula and other mathematical concepts. Choosing the right method depends on the specific equation you're dealing with. For some equations, factoring might be the most efficient approach. For others, the quadratic formula might be the only option. As you practice, you'll develop a sense for which method is best suited for different situations. Each method provides a unique lens through which to view and understand quadratic equations. This multifaceted approach not only equips you with problem-solving tools but also deepens your comprehension of the underlying mathematical principles. So, let's dive into applying these methods to our equation and see which one leads us to the solution.

Solving by Factoring

Let's start by trying the factoring method. Factoring is often the quickest way to solve a quadratic equation if it's possible. Remember, factoring means rewriting the quadratic expression as a product of two binomials. So, we want to find two expressions that, when multiplied together, give us x² + 12x + 36. Here's how we can approach it. We need to find two numbers that add up to 12 (the coefficient of the x term) and multiply to 36 (the constant term). Think of pairs of numbers that multiply to 36: 1 and 36, 2 and 18, 3 and 12, 4 and 9, and 6 and 6. Which of these pairs also adds up to 12? You got it – 6 and 6. This means we can factor the quadratic expression as (x + 6)(x + 6). Now we have (x + 6)(x + 6) = 0. For the product of two factors to be zero, at least one of them must be zero. In this case, both factors are the same, so we just need to solve x + 6 = 0. Subtracting 6 from both sides gives us x = -6. And there you have it! We've found one solution by factoring. Since both factors were the same, this equation has a single, repeated root. Factoring is like cracking a code – once you find the right combination, the solution unfolds smoothly. This method highlights the power of recognizing patterns and relationships within the equation, making the solution process both efficient and insightful. So, by successfully factoring our equation, we've not only found the value of x but also reinforced our understanding of how quadratic expressions can be manipulated to reveal their hidden solutions.

The Quadratic Formula Approach

Now, let's tackle this problem using the quadratic formula. Even though we already found the solution by factoring, it's great to see how the quadratic formula works. It's a powerful tool that can solve any quadratic equation, even those that don't factor easily. Remember the quadratic formula? It's x = [-b ± √(b² - 4ac)] / 2a. Don't let it scare you! It's just a formula where we plug in the values of a, b, and c from our equation. In our case, a = 1, b = 12, and c = 36. Let's plug these values into the formula: x = [-12 ± √(12² - 4 * 1 * 36)] / (2 * 1). Now, let's simplify. First, calculate the expression under the square root: 12² - 4 * 1 * 36 = 144 - 144 = 0. This means the square root part simplifies to √0, which is just 0. So, our equation becomes x = [-12 ± 0] / 2. This simplifies to x = -12 / 2, which gives us x = -6. See? We got the same answer as we did by factoring! The quadratic formula gives us the solution directly, without the need to find factors. It's like having a universal key that unlocks the solution to any quadratic equation. While it might seem more complex than factoring at first, it's a reliable method that always works. This consistency makes the quadratic formula an indispensable tool in your mathematical arsenal. By applying it to our equation, we've not only confirmed our previous solution but also gained confidence in using this powerful formula for future challenges. So, whether factoring is straightforward or not, the quadratic formula stands ready to provide the answer.

Analyzing the Solutions and the Discriminant

We've found that x = -6 is the solution to our quadratic equation. But let's take a moment to dig a little deeper and understand why we got this specific solution and what it tells us about the equation. Remember that part of the quadratic formula under the square root, b² - 4ac? This is called the discriminant, and it's a crucial piece of information. The discriminant tells us about the nature of the solutions. If the discriminant is positive, the equation has two distinct real solutions. If it's zero, the equation has one real solution (a repeated root). And if it's negative, the equation has no real solutions (it has two complex solutions, but we won't get into those right now). In our case, the discriminant was 12² - 4 * 1 * 36 = 0. This confirms that we have one real solution, which we found to be x = -6. The fact that we have a repeated root means the graph of the quadratic equation (a parabola) touches the x-axis at only one point. It's like the parabola kisses the x-axis instead of crossing it. Understanding the discriminant gives us a deeper insight into the behavior of quadratic equations. It's not just about finding the solutions; it's about understanding what those solutions mean in the context of the equation and its graph. This analytical perspective transforms problem-solving from a mechanical process into a meaningful exploration of mathematical concepts. By examining the discriminant, we gain a more complete picture of the quadratic equation, appreciating its characteristics and the nature of its solutions. This deeper understanding empowers us to tackle a wider range of problems with greater confidence and insight.

Choosing the Correct Alternative

Alright, we've cracked the code! We know that x = -6 is the solution to the quadratic equation x² + 12x + 36 = 0. Now, let's look at the alternatives provided and choose the correct one. The alternatives were: A) -6, B) 6, C) -12, D) 12. Based on our calculations, the correct answer is A) -6. We arrived at this solution using both factoring and the quadratic formula, so we can be confident in our answer. It's always a good idea to double-check your work, especially when you have multiple methods to verify your solution. In this case, we used two different approaches and arrived at the same answer, which strengthens our confidence. When faced with multiple-choice questions, it's helpful to not only find the correct answer but also understand why the other options are incorrect. This reinforces your understanding of the concept and helps you avoid common mistakes. Choosing the correct alternative is the final step in the problem-solving process, but it's just as important as the earlier steps. It demonstrates your ability to not only solve the problem but also to interpret the results and apply them to the given context. So, with a clear understanding of the solution and the process, we can confidently select the correct answer and move on to the next challenge. This methodical approach ensures accuracy and builds a solid foundation for future mathematical endeavors.

Conclusion

So, there you have it! We've successfully found the value of x in the quadratic equation x² + 12x + 36 = 0. We used factoring and the quadratic formula, and we even explored the discriminant to understand the nature of the solutions. The answer, as we've confirmed, is x = -6. Solving quadratic equations is a fundamental skill in algebra, and it's one that you'll use in many different contexts. Whether you're solving physics problems, designing structures, or even working with computer graphics, quadratic equations often pop up. By mastering these techniques, you're building a solid foundation for more advanced math and science. Remember, practice makes perfect. The more you solve quadratic equations, the more comfortable you'll become with the different methods and the faster you'll be able to find the solutions. Don't be afraid to try different approaches and to make mistakes along the way. Mistakes are learning opportunities! Keep exploring, keep questioning, and keep solving. Math is a journey, and every problem you solve is a step forward. So, keep up the great work, and I'll see you in the next math adventure!

Keywords: quadratic equation, factoring, quadratic formula, discriminant, solve for x

Alternatives: A) -6, B) 6, C) -12, D) 12