Solving X In X² - 10x + 25 = 0 A Step-by-Step Guide
Hey guys! Let's dive into the fascinating world of quadratic equations. Today, we're going to tackle a classic example: x² - 10x + 25 = 0. Don't worry, it might look intimidating at first, but we'll break it down step by step so everyone can follow along. Quadratic equations are fundamental in mathematics, appearing in various fields like physics, engineering, and even economics. Understanding how to solve them is a crucial skill for any student or anyone interested in problem-solving. We'll explore different methods to find the value(s) of x that satisfy this equation. So, grab your pencils and notebooks, and let's get started on this mathematical adventure! We'll cover everything from factoring to using the quadratic formula, ensuring you have a solid grasp on solving quadratic equations. Remember, practice makes perfect, so the more you work through these problems, the more confident you'll become. We aim to make this guide super clear and helpful, so you can confidently solve similar equations in the future. Let's make math fun and accessible together!
Understanding Quadratic Equations
Before we jump into solving our specific equation, let's take a moment to understand what a quadratic equation actually is. A quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (in our case, 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. If a were 0, the equation would become a linear equation, not a quadratic one. The coefficients a, b, and c play a crucial role in determining the nature and solutions of the quadratic equation. Understanding this general form is essential because it allows us to recognize quadratic equations in various contexts and apply appropriate solution methods. In our example, x² - 10x + 25 = 0, we can identify a as 1, b as -10, and c as 25. These values will be important when we use methods like the quadratic formula. Recognizing the coefficients correctly is the first step in solving any quadratic equation. Think of a, b, and c as the building blocks of the equation; they define its shape and behavior. Now that we have a solid understanding of the general form, let's move on to the first method for solving our equation: factoring.
Method 1 Factoring the Quadratic Equation
Factoring is often the quickest and most straightforward method for solving quadratic equations, if the equation can be factored easily. The goal of factoring is to rewrite the quadratic expression as a product of two binomials. For our equation, x² - 10x + 25 = 0, we're looking for two numbers that multiply to c (25) and add up to b (-10). Think of it like a puzzle where we need to find the right pieces that fit together. In this case, the numbers are -5 and -5 because (-5) * (-5) = 25 and (-5) + (-5) = -10. Once we've found these numbers, we can rewrite the equation in factored form: (x - 5)(x - 5) = 0. Notice how the -5s directly appear in the binomials. This factored form is equivalent to the original quadratic equation but is much easier to solve. Now, we use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. So, we set each factor equal to zero: x - 5 = 0. Solving this simple linear equation gives us x = 5. Since both factors are the same, we only get one solution. This means our quadratic equation has a repeated root. Factoring is a powerful tool, but it's essential to remember that not all quadratic equations can be easily factored. In those cases, we'll need to turn to other methods, such as the quadratic formula, which we'll discuss later. But for now, let's appreciate the elegance and efficiency of factoring when it works!
Method 2 Using the Quadratic Formula
When factoring isn't straightforward, or if you just prefer a more general approach, the quadratic formula is your best friend. The quadratic formula is a universal method for solving any quadratic equation of the form ax² + bx + c = 0. It provides the solution(s) for x using the coefficients a, b, and c. The formula is given by: x = (-b ± √(b² - 4ac)) / (2a). It might look a bit scary at first, but trust me, it's a powerful tool once you get the hang of it. Let's apply the quadratic formula to our equation, x² - 10x + 25 = 0. We identified earlier that a = 1, b = -10, and c = 25. Now, we simply plug these values into the formula: x = (-(-10) ± √((-10)² - 4 * 1 * 25)) / (2 * 1). Simplifying this expression step by step, we get: x = (10 ± √(100 - 100)) / 2. Further simplification yields: x = (10 ± √0) / 2. Since the square root of 0 is 0, we have: x = 10 / 2, which gives us x = 5. As we found with factoring, the quadratic formula also gives us a single solution, x = 5. The term inside the square root, b² - 4ac, is called the discriminant. The discriminant tells us about the nature of the solutions. If it's positive, there are two distinct real solutions; if it's zero, there is one real solution (a repeated root); and if it's negative, there are two complex solutions. In our case, the discriminant is 0, confirming that we have one real solution. The quadratic formula is a reliable method that works for all quadratic equations, making it an essential tool in your mathematical toolkit.
Verifying the Solution
After solving a quadratic equation, it's always a good idea to verify your solution. This helps ensure that you haven't made any mistakes in your calculations. To verify our solution x = 5 for the equation x² - 10x + 25 = 0, we simply substitute x = 5 back into the original equation: (5)² - 10(5) + 25 = 0. Evaluating this expression, we get: 25 - 50 + 25 = 0, which simplifies to 0 = 0. Since the equation holds true, our solution x = 5 is indeed correct. Verification is a crucial step in problem-solving. It not only confirms the correctness of your answer but also helps reinforce your understanding of the equation and its solution. It's like a final checkmark that gives you confidence in your work. In exams or real-world applications, verifying your solution can save you from costly errors. So, always take a moment to plug your solution back into the original equation and ensure everything checks out. This simple step can make a big difference in the accuracy and reliability of your results. We've now successfully solved and verified our quadratic equation, demonstrating both factoring and the quadratic formula. Let's recap what we've learned.
Conclusion and Key Takeaways
Alright, guys, we've reached the end of our journey to solve the quadratic equation x² - 10x + 25 = 0. We've explored two main methods: factoring and using the quadratic formula. Factoring allowed us to rewrite the equation as (x - 5)(x - 5) = 0, leading to the solution x = 5. The quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), provided the same solution, reinforcing its reliability as a universal method. We also verified our solution by substituting x = 5 back into the original equation, confirming its correctness. The key takeaway here is that quadratic equations can be solved in multiple ways, and choosing the most efficient method depends on the specific equation. Factoring is often quicker when the equation is easily factorable, while the quadratic formula works for all quadratic equations. Understanding the discriminant, b² - 4ac, helps us determine the nature of the solutions. A discriminant of 0 indicates one real solution, as we saw in our example. Mastering these techniques is crucial for success in algebra and beyond. Quadratic equations are fundamental building blocks in many mathematical and scientific applications. So, keep practicing, and don't be afraid to tackle more complex problems. Remember, the more you practice, the more confident and proficient you'll become. Keep up the great work, and happy solving!
Further Practice
To really solidify your understanding of solving quadratic equations, it's essential to practice with a variety of examples. Let's consider a few more quadratic equations you can try solving on your own. This hands-on practice will help you become more comfortable with both factoring and using the quadratic formula. Try these equations:
- x² - 4x + 4 = 0
- 2x² + 5x - 3 = 0
- x² + 6x + 9 = 0
- 3x² - 10x + 3 = 0
For each equation, try factoring first. If factoring seems difficult, then apply the quadratic formula. Remember to identify a, b, and c correctly before plugging them into the formula. After you find the solution(s), don't forget to verify them by substituting them back into the original equation. This practice will help you develop a strong intuition for solving quadratic equations and recognizing patterns. If you encounter any difficulties, don't hesitate to review the methods we discussed earlier or seek help from online resources or your instructor. Learning math is like building a puzzle – each piece you learn makes the bigger picture clearer. So, keep practicing and enjoy the process of mastering quadratic equations! And always remember, the more you practice, the easier it gets. Solving these equations will not only improve your math skills but also boost your confidence in tackling more challenging problems in the future. Keep up the great work, and you'll be a quadratic equation expert in no time!
