Solving X² - 22x + 121 = 0 A Step By Step Guide
Hey there, math enthusiasts! Ever stumbled upon a quadratic equation that looks like a puzzle? Well, you're not alone. Quadratic equations are a fundamental part of algebra, and mastering them opens doors to more advanced mathematical concepts. Today, we're diving deep into how to solve one such equation: x² - 22x + 121 = 0. Buckle up, because we're about to break it down step-by-step, making it super easy to understand.
Understanding Quadratic Equations
Before we jump into solving, let's quickly recap what quadratic equations are all about. A quadratic equation is a polynomial equation of the second degree. The general form looks like this: ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is the variable we're trying to solve for. The solutions to a quadratic equation are also known as the roots or zeros of the equation. These are the values of 'x' that make the equation true. Solving quadratic equations is a crucial skill in various fields, including physics, engineering, and computer science. Whether you're calculating trajectories, designing structures, or developing algorithms, understanding quadratic equations is a must. So, let's make sure we've got a solid grasp on the fundamentals before we move on. Quadratic equations aren't just abstract mathematical concepts; they have real-world applications all around us. From the curve of a projectile's flight path to the shape of a satellite dish, quadratic equations help us model and understand the world. Now that we've got the basics covered, let's dive into the heart of the matter: solving the equation x² - 22x + 121 = 0. We'll explore different methods and techniques, so you can choose the one that works best for you. Get ready to put your problem-solving skills to the test!
Identifying the Key Components
First things first, let's identify the components of our equation, x² - 22x + 121 = 0. We can see that 'a' (the coefficient of x²) is 1, 'b' (the coefficient of x) is -22, and 'c' (the constant term) is 121. Recognizing these components is the first step toward choosing the right method to solve the equation. In this case, we'll focus on factoring, a powerful technique that's particularly effective when the quadratic equation can be expressed as a product of two binomials. Factoring is like reverse-engineering the multiplication process. Instead of multiplying binomials to get a quadratic equation, we're trying to break down the quadratic equation back into its binomial factors. This method is elegant and efficient, but it requires a keen eye for patterns and relationships between the coefficients. Don't worry if it seems a bit tricky at first. With practice, you'll become a factoring pro! Now, let's roll up our sleeves and start the factoring process. We'll be looking for two numbers that add up to 'b' (-22) and multiply to 'c' (121). Think you can spot them? Keep reading to find out!
Method 1: Factoring the Quadratic Equation
Factoring is often the quickest way to solve a quadratic equation, especially when the equation can be easily factored. The goal here is to rewrite the quadratic expression as a product of two binomials. This method hinges on finding two numbers that satisfy specific conditions related to the coefficients of the quadratic equation. When factoring, we need to find two numbers that multiply to the constant term (c) and add up to the coefficient of the linear term (b). In our equation, x² - 22x + 121 = 0, we need two numbers that multiply to 121 and add up to -22. Can you think of any numbers that fit the bill? The numbers -11 and -11 perfectly fit these conditions because (-11) * (-11) = 121 and (-11) + (-11) = -22. Once we've identified these numbers, we can rewrite the quadratic equation in factored form. This transformation is the key to unlocking the solutions of the equation. By expressing the quadratic as a product of binomials, we set the stage for applying 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. This principle allows us to break down the problem into simpler equations, each of which can be solved independently. Factoring is not just a mechanical process; it's an art that requires practice and intuition. The more you factor quadratic equations, the better you'll become at spotting patterns and relationships between the coefficients. So, don't be discouraged if you don't get it right away. Keep practicing, and you'll soon master this powerful technique.
Finding the Factors
As we discussed, we need to find two numbers that multiply to 121 and add up to -22. The numbers -11 and -11 fit the bill perfectly! This means we can rewrite our equation as: (x - 11)(x - 11) = 0. Notice how the numbers we found directly translate into the terms within the binomial factors. This is a crucial step in the factoring process. It's like fitting puzzle pieces together to reveal the underlying structure of the equation. Now that we have the equation in factored form, we're ready to apply the zero-product property. This property is the key to unlocking the solutions of the quadratic equation. The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. In our case, we have two factors: (x - 11) and (x - 11). The product of these factors is zero, so we can set each factor equal to zero and solve for 'x'. This breaks the problem down into two simpler equations, each of which is easy to solve. By applying the zero-product property, we transform the quadratic equation into a set of linear equations. This is a common strategy in mathematics: breaking down complex problems into simpler ones. Now, let's take the next step and apply the zero-product property to find the solutions of our equation. We're almost there!
Applying the Zero-Product Property
The zero-product property states that if ab = 0, then a = 0 or b = 0 (or both). Applying this to our factored equation (x - 11)(x - 11) = 0, we get: x - 11 = 0. Since both factors are the same, we only need to solve this equation once. This equation is now a simple linear equation, which is much easier to solve than the original quadratic equation. We've successfully transformed a complex problem into a simple one. By applying the zero-product property, we've isolated the variable 'x' and set the stage for finding its value. Solving linear equations is a fundamental skill in algebra, and it's essential for solving quadratic equations by factoring. Now, let's take the final step and solve for 'x'. We're almost at the finish line!
Solving for x
Adding 11 to both sides of the equation x - 11 = 0, we find that x = 11. So, the solution to the equation x² - 22x + 121 = 0 is x = 11. But wait, there's more to the story! Notice that we have a repeated factor (x - 11). This means that x = 11 is a repeated root, or a root with multiplicity 2. A repeated root indicates that the parabola represented by the quadratic equation touches the x-axis at only one point, instead of crossing it at two distinct points. Understanding the concept of repeated roots is crucial for interpreting the solutions of quadratic equations and their graphical representations. In our case, the parabola touches the x-axis at x = 11, indicating that this is the only solution to the equation. So, not only have we found the solution, but we've also gained insights into the nature of the roots and the behavior of the quadratic function. Congratulations, you've successfully solved the quadratic equation by factoring! Now, let's summarize our findings and express the solution set.
Solution Set
The solution set for the equation x² - 22x + 121 = 0 is {11}. Since we have a repeated root, we only include the solution once in the set. This concise notation neatly encapsulates all the solutions to the equation. The solution set is a fundamental concept in mathematics, as it provides a clear and unambiguous way to represent the values that satisfy an equation. In our case, the solution set {11} tells us that the only value of 'x' that makes the equation true is 11. Understanding and expressing solution sets is crucial for communicating mathematical ideas effectively. It's like a mathematical shorthand that allows us to convey complex information in a compact and precise way. Now that we've found the solution set, let's take a moment to reflect on the process we followed. We started by understanding the basics of quadratic equations, then identified the key components of our equation, and finally applied the factoring method to find the solution. This step-by-step approach is a valuable strategy for tackling any mathematical problem. Congratulations, you've mastered the art of solving quadratic equations by factoring! Now, go forth and conquer more mathematical challenges!
Method 2: Using the Quadratic Formula
If factoring seems tricky, don't worry! There's another powerful tool in our arsenal: the quadratic formula. This formula is a universal solution for any quadratic equation, regardless of whether it can be easily factored or not. The quadratic formula is derived from the process of completing the square, and it provides a direct way to calculate the roots of a quadratic equation. It's like a magic wand that transforms any quadratic equation into its solutions. The formula is given by: x = (-b ± √(b² - 4ac)) / 2a. This formula might look intimidating at first, but it's actually quite straightforward once you understand its components. 'a', 'b', and 'c' are the coefficients of the quadratic equation, just like we identified earlier. The ± symbol indicates that there are two possible solutions: one with a plus sign and one with a minus sign. The square root part, √(b² - 4ac), is known as the discriminant. The discriminant tells us about the nature of the roots: whether they are real or complex, and whether they are distinct or repeated. The quadratic formula is a powerful tool, but it's important to use it correctly. Make sure you substitute the values of 'a', 'b', and 'c' carefully, and pay attention to the order of operations. With practice, you'll become a master of the quadratic formula. Now, let's apply this formula to our equation and see how it works!
Applying the Formula
For our equation, x² - 22x + 121 = 0, we have a = 1, b = -22, and c = 121. Plugging these values into the quadratic formula, we get: x = (22 ± √((-22)² - 4 * 1 * 121)) / (2 * 1). See how we carefully substituted the values into their respective places? This is crucial for getting the correct answer. Now, let's simplify the expression step-by-step. We'll start by calculating the discriminant, which is the part under the square root. This calculation will tell us about the nature of the roots: whether they are real or complex, and whether they are distinct or repeated. Simplifying the expression is like peeling away the layers of an onion. We start with the innermost operations and work our way outwards. This systematic approach ensures that we don't make any mistakes along the way. Don't be intimidated by the complexity of the formula. Just take it one step at a time, and you'll be amazed at how easily it unravels. Now, let's continue simplifying and see what we get!
Simplifying the Expression
Let's simplify! First, calculate the discriminant: (-22)² - 4 * 1 * 121 = 484 - 484 = 0. Aha! The discriminant is 0. This tells us that the equation has exactly one real solution (a repeated root). A discriminant of zero is a special case that indicates a perfect square trinomial. This means that the quadratic equation can be factored into a binomial squared, which we already discovered when we factored the equation earlier. The discriminant is like a mathematical detective that provides clues about the nature of the roots. It helps us understand the behavior of the quadratic function and its graphical representation. Now that we know the discriminant is zero, we can simplify the quadratic formula further. The square root of zero is zero, so the ± part of the formula disappears, leaving us with a single solution. Let's continue simplifying and find that solution!
Finding the Solution
Since the square root part is 0, our equation simplifies to: x = 22 / 2 = 11. So, we arrive at the same solution as before: x = 11. The quadratic formula confirms our earlier result from factoring. This is a great way to check your work and ensure that you've found the correct solution. The quadratic formula is a powerful tool, but it's not a substitute for understanding the underlying concepts. It's important to know when to use the formula and how to interpret the results. In this case, the quadratic formula provided us with the same solution we found by factoring, reinforcing our understanding of the equation. Now that we've solved the equation using two different methods, we can be confident in our answer. Let's summarize our findings and express the solution set once again.
Conclusion
In this comprehensive guide, we've explored how to find the solution set for the quadratic equation x² - 22x + 121 = 0. We used two methods: factoring and the quadratic formula, both leading us to the same solution, x = 11. The solution set is {11}, indicating a repeated root. We've not only found the solution but also gained a deeper understanding of quadratic equations and the techniques for solving them. Quadratic equations are a fundamental part of mathematics, and mastering them opens doors to more advanced concepts. Whether you're solving real-world problems or exploring abstract mathematical ideas, the skills you've learned here will serve you well. Remember, practice makes perfect. The more you work with quadratic equations, the more confident and proficient you'll become. So, keep solving, keep exploring, and keep learning! Congratulations on mastering the art of solving quadratic equations!