Solving (x-7)^2 = 9: Step-by-Step Solutions

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Hey guys! Let's dive into solving this equation: (x-7)^2 = 9. If you're scratching your head trying to figure out where to start, don't worry! We're going to break it down step by step so you can ace problems like this. Math can be intimidating, but with a little patience and the right approach, you'll be solving quadratic equations in no time.

Understanding the Problem

Before we jump into the solution, let's make sure we understand what the question is asking. We need to find the values of x that make the equation (x-7)^2 = 9 true. This is a quadratic equation, and quadratic equations often have two solutions. That's why you see answer choices with pairs of numbers. So, our mission is to find those two special values of x. Identifying the type of equation is the first key step in solving it. Recognizing it as a quadratic equation helps us choose the appropriate method. Now, let's explore the different methods we can use to tackle this problem. We want to choose the most efficient and accurate method to arrive at the correct solutions. Are you ready to put on your math hats and get started? Let’s begin with the first approach and unlock the mysteries of this equation together!

Method 1: Taking the Square Root

The most straightforward way to solve this equation is by taking the square root of both sides. This method works beautifully when you have a perfect square isolated on one side of the equation, which is exactly what we have here. Remember, when you take the square root, you need to consider both the positive and negative roots. This is crucial because both the positive and negative values, when squared, will give you the same positive result. Let's see how it works:

  1. Start with the equation: (x-7)^2 = 9
  2. Take the square root of both sides: √(x-7)^2 = ±√9
  3. Simplify: x - 7 = ±3

Now we have two separate equations to solve:

  • x - 7 = 3
  • x - 7 = -3

Let's solve the first one. To isolate x, we add 7 to both sides of the equation. This gives us x = 3 + 7, which simplifies to x = 10. Great! We've found one solution. Now, let’s tackle the second equation. Again, we want to get x by itself, so we add 7 to both sides of x - 7 = -3. This results in x = -3 + 7, which simplifies to x = 4. Awesome! We've found our second solution.

So, the solutions are x = 10 and x = 4. This matches option (D) in the answer choices. This method highlights the importance of remembering both positive and negative roots when dealing with square roots in equations. Forgetting the negative root is a common mistake, so always double-check! Now, let's explore another method to solve the same equation. This will give you a broader perspective and help you choose the method that best suits your style and the problem at hand. Are you curious to see how else we can crack this equation? Let's move on to Method 2!

Method 2: Expanding and Factoring

Another way to solve this equation is by expanding the square, rearranging the terms, and then factoring the resulting quadratic. This method is a bit more involved than taking the square root directly, but it's a valuable technique to have in your problem-solving toolkit. It can be particularly useful when the equation isn't set up as a perfect square directly. Here's how it works:

  1. Start with the equation: (x-7)^2 = 9
  2. Expand the left side: (x-7)(x-7) = x^2 - 14x + 49
  3. So now we have: x^2 - 14x + 49 = 9
  4. Subtract 9 from both sides to set the equation to zero: x^2 - 14x + 40 = 0

Now we have a standard quadratic equation in the form ax^2 + bx + c = 0. To solve this, we need to factor the quadratic expression. Factoring involves finding two numbers that multiply to give the constant term (40 in this case) and add up to the coefficient of the x term (-14). Let’s think about the factors of 40: 1 and 40, 2 and 20, 4 and 10, 5 and 8. The pair 4 and 10 looks promising. Since we need them to add up to -14, we'll use -4 and -10.

So, we can factor the quadratic as: (x - 4)(x - 10) = 0

Now, for this product to be zero, at least one of the factors must be zero. This gives us two possible equations:

  • x - 4 = 0
  • x - 10 = 0

Solving these equations is simple. For the first equation, add 4 to both sides to get x = 4. For the second equation, add 10 to both sides to get x = 10. Just like that, we've found the same solutions as before: x = 4 and x = 10. This confirms our answer and showcases the versatility of different problem-solving methods. Expanding and factoring might seem longer, but it reinforces key algebraic skills. It also demonstrates how different approaches can lead to the same correct answer, which is a comforting thought when you're tackling tough problems. Let's recap our solutions and then talk about why this works!

The Solutions

Both methods led us to the same solutions: x = 4 and x = 10. This corresponds to answer choice (D). It’s always reassuring when different methods converge on the same answer. It boosts our confidence in the correctness of the solution. Plus, seeing multiple approaches reinforces our understanding of the underlying concepts. These solutions tell us that if we substitute either 4 or 10 for x in the original equation, the equation will hold true. This is the essence of solving equations – finding the values that satisfy the given relationship.

Now, let’s take a step back and think about why these methods work. Understanding the “why” behind the math is just as important as knowing the “how.” When we took the square root, we were essentially undoing the squaring operation. When we expanded and factored, we were using 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. These are fundamental principles in algebra, and mastering them will serve you well in more advanced math courses. So, keep practicing, keep exploring different methods, and always remember to ask “why.” That curiosity is the key to unlocking deeper mathematical understanding. Next, let’s summarize the key takeaways from this problem-solving journey!

Key Takeaways

  • Recognize the equation type: Identifying the equation as quadratic helped us choose appropriate solution methods.
  • Taking the square root: This method is efficient when you have a perfect square isolated.
  • Remember both roots: Don't forget the positive and negative square roots!
  • Expanding and factoring: This method works for any quadratic equation.
  • Zero-product property: A crucial principle when factoring.
  • Multiple methods: There's often more than one way to solve a problem.

By understanding these key takeaways, you'll be better equipped to tackle similar problems in the future. Each problem you solve adds another tool to your mathematical toolbox. Keep practicing, and you'll become a confident problem solver! Remember, math isn't just about finding the right answer; it's about the journey of understanding and the skills you develop along the way. So, embrace the challenge, enjoy the process, and keep learning! Now you know how to solve equations like (x-7)^2 = 9. You've got this!