Solving (x-5)(x-5)=0 Find The Solution Set

Hey guys! Let's dive into solving the equation (x-5)(x-5) = 0. This might look a bit intimidating at first, but trust me, it's totally manageable. We're going to break it down step by step, so by the end of this article, you'll be a pro at solving similar equations. We'll not only find the solution set but also understand the underlying concepts, ensuring you're well-equipped to tackle more complex problems in the future. So, grab your thinking caps, and let's get started!

Understanding the Basics: What is a Solution Set?

Before we jump into the equation itself, let's quickly recap what a solution set actually means. In simple terms, a solution set is the collection of all values that, when substituted for the variable (in our case, x), make the equation true. Think of it like a puzzle – we're trying to find the specific pieces (x values) that fit perfectly and complete the picture (the equation).

In the equation (x-5)(x-5) = 0, we're looking for the value(s) of x that, when plugged in, will make the entire expression equal to zero. This is a fundamental concept in algebra, and understanding it is crucial for solving various types of equations, from simple linear equations to more complex polynomial equations. The beauty of mathematics lies in its consistency, and this principle of finding values that satisfy an equation is a cornerstone of algebraic problem-solving. So, with this understanding in mind, let's move on to the specific methods we can use to crack this equation.

Method 1: The Zero Product Property

The Zero Product Property is our best friend when dealing with equations like this one. This property states that if the product of two or more factors is equal to zero, then at least one of the factors must be zero. In mathematical terms, if a * b = 0*, then either a = 0 or b = 0 (or both!). This seemingly simple rule is incredibly powerful and forms the basis for solving many algebraic equations. It allows us to break down a complex equation into simpler parts, making it much easier to handle.

In our equation, (x-5)(x-5) = 0, we have two factors: (x-5) and (x-5). Applying the Zero Product Property, we can say that for the entire equation to be zero, at least one of these factors must be zero. This leads us to a much simpler equation to solve: x - 5 = 0. This is where the magic happens – we've transformed a potentially daunting problem into a straightforward one. Now, we just need to isolate x and find its value. By adding 5 to both sides of the equation, we get x = 5. This is our solution! But wait, there's more to explore in the next sections.

Method 2: Expanding and Factoring (Optional, but Insightful)

While the Zero Product Property is the most direct route for this particular equation, it's always good to have more tools in your toolbox. Let's explore an alternative method: expanding and factoring. This method might seem a bit longer for this specific problem, but it's a valuable technique for solving quadratic equations in general.

First, we expand the equation (x-5)(x-5) = 0. Remember, expanding means multiplying out the factors. In this case, we're multiplying (x-5) by itself. Using the distributive property (or the FOIL method), we get:

(x - 5)(x - 5) = xx - 5x - 5x + 25 = x^2 - 10x + 25*

So, our equation now looks like: x^2 - 10x + 25 = 0. This is a quadratic equation, which is an equation of the form ax^2 + bx + c = 0. Now, we need to factor this quadratic expression. Factoring is the reverse of expanding – we're trying to find the factors that multiply together to give us the quadratic expression. In this case, we're looking for two numbers that multiply to 25 and add up to -10. Those numbers are -5 and -5. Therefore, we can factor the quadratic expression as:

x^2 - 10x + 25 = (x - 5)(x - 5)

Notice anything familiar? We're back to our original factored form! This confirms that our expansion and factoring were done correctly. Now, we can apply the Zero Product Property as before, leading us to the same solution: x = 5. While this method might seem like a detour for this specific problem, it's a crucial skill for solving other quadratic equations where the factoring might not be as obvious. Understanding both methods gives you a more comprehensive understanding of algebraic problem-solving.

The Solution Set: x = 5

Alright, we've crunched the numbers and explored different methods, and the result is clear: the solution set for the equation (x-5)(x-5) = 0 is x = 5. But what does this mean in the grand scheme of things? Well, it means that 5 is the only value that, when substituted for x in the original equation, will make the equation true. You can try it out yourself! Plug in 5 for x and you'll see that (5-5)(5-5) = 0 * 0 = 0*. It works like a charm!

It's also important to note that this equation has a repeated root. A repeated root occurs when the same factor appears multiple times in the equation. In this case, the factor (x-5) appears twice. This means that the solution x = 5 is a root of multiplicity 2. This concept is important in higher-level mathematics, especially when dealing with polynomial functions and their graphs. The multiplicity of a root affects the behavior of the graph at that point. So, understanding repeated roots is a valuable step in your mathematical journey.

Why is this important? Real-world applications

You might be thinking, "Okay, I can solve this equation, but why is it important? Where will I ever use this in real life?" That's a valid question! While you might not encounter this exact equation in your daily life, the underlying principles are used extensively in various fields. Solving equations is a fundamental skill in mathematics and is crucial for problem-solving in many different areas.

For example, engineers use equations to design structures, predict the behavior of circuits, and optimize systems. Physicists use equations to model the motion of objects, understand the behavior of particles, and explore the mysteries of the universe. Economists use equations to analyze markets, predict economic trends, and develop financial models. Even computer scientists rely on equations to develop algorithms, design software, and solve complex computational problems. The applications are endless!

The ability to solve equations, including quadratic equations like the one we tackled today, is a building block for more advanced mathematical concepts. It's like learning the alphabet before you can write a novel. Mastering these fundamental skills will open doors to a wide range of opportunities in STEM fields and beyond. So, keep practicing, keep exploring, and keep building your mathematical foundation!

Conclusion: Mastering the Basics

So there you have it, guys! We've successfully solved the equation (x-5)(x-5) = 0 and found the solution set: x = 5. We explored different methods, understood the Zero Product Property, and even delved into the concept of repeated roots. More importantly, we discussed why these skills are valuable and how they apply to real-world scenarios. Remember, mathematics is not just about memorizing formulas; it's about understanding concepts and developing problem-solving skills.

The key takeaway here is the importance of mastering the basics. A solid foundation in algebra will make your journey through more advanced mathematical topics much smoother. Don't be afraid to ask questions, seek help when needed, and practice regularly. The more you practice, the more confident and proficient you'll become. Keep exploring the fascinating world of mathematics, and you'll be amazed at what you can achieve! Solving equations is just the beginning – there's a whole universe of mathematical concepts waiting to be discovered. Keep learning, keep growing, and never stop questioning!