Solving $(x-5)^2=49$ By The Square Root Property A Step-by-Step Guide

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Hey there, math enthusiasts! Today, we're diving into a super cool technique for solving quadratic equations: the square root property. This method is especially handy when you've got a squared term isolated on one side of the equation. Let's break it down and get you solving like a pro!

What is the Square Root Property?

The square root property, guys, is all about undoing a square. Think of it like this: if you have something squared equal to a number, you can take the square root of both sides to get rid of the square. The key thing to remember is that when you take the square root, you need to consider both the positive and negative roots. This is because both a positive number and its negative counterpart, when squared, give you a positive result. For example, both 5² and (-5)² equal 25. This principle is the core of the square root property, and understanding it is crucial for successfully applying the technique. The square root property is not just a shortcut; it’s a fundamental concept that stems from the definition of square roots and squares. A square root of a number n is a value that, when multiplied by itself, gives n. Since both a positive and a negative value can satisfy this condition for any positive n, we must consider both possibilities when solving equations. Ignoring the negative root is a common mistake that can lead to incomplete or incorrect solutions. By remembering to include both roots, we ensure that we capture all possible solutions to the equation, providing a comprehensive and accurate answer. This thoroughness is what makes the square root property a reliable and powerful tool in algebra. Moreover, the square root property is deeply connected to the quadratic formula, which is a general method for solving quadratic equations. While the quadratic formula can be applied to any quadratic equation, the square root property offers a more direct and efficient approach when the equation is in a specific form, namely when a squared expression is isolated. Recognizing when to use the square root property can save time and effort, making it an invaluable technique in your mathematical toolkit. It's also worth noting that the square root property is not limited to simple numerical values. It can also be applied to more complex expressions and equations involving variables and other algebraic terms. The underlying principle remains the same: to undo a square, take the square root of both sides, remembering to account for both positive and negative roots. By mastering this concept, you'll be well-equipped to tackle a wide range of mathematical problems, from basic algebra to more advanced topics.

How to Apply the Square Root Property: A Step-by-Step Guide

Alright, let's get down to the nitty-gritty. How do we actually use this property to solve equations? It's simpler than it sounds, trust me! Follow these steps, and you'll be golden:

  1. Isolate the squared term: First things first, you need to get the term that's being squared all by itself on one side of the equation. This might involve adding, subtracting, multiplying, or dividing. Think of it as setting the stage for the main event – taking the square root.
  2. Take the square root of both sides: Once you've isolated the squared term, take the square root of both sides of the equation. Remember the crucial part: include both the positive and negative square roots. This is where that ± symbol comes into play. This step is the heart of the square root property, where we undo the square and reveal the possible values of the variable. It's a direct application of the inverse operation, allowing us to move closer to the solution. Forgetting to include both the positive and negative roots is a common mistake, so always double-check to ensure you've accounted for all possibilities.
  3. Solve for the variable: After taking the square root, you'll likely have two separate equations to solve, one with the positive root and one with the negative root. Solve each of these equations to find the possible values of your variable. This might involve some basic algebraic manipulations, such as adding, subtracting, multiplying, or dividing. It's important to handle each equation carefully and methodically to avoid errors. The goal is to isolate the variable completely, revealing its possible values that satisfy the original equation. This step is where you put all the pieces together, transforming the square root expressions into clear, numerical solutions. And this is the part where you get to see the fruits of your labor. Completing this step accurately ensures you have captured all the solutions to the equation, leaving no possibilities unexplored. It's a satisfying moment when you see the variable standing alone, its value revealed.
  4. Check your solutions: It's always a good idea to plug your solutions back into the original equation to make sure they work. This helps you catch any mistakes you might have made along the way. It’s like proofreading your work or double-checking your calculations to ensure accuracy. This step is a crucial safety net in mathematics, preventing you from confidently submitting a wrong answer. By substituting your solutions back into the original equation, you can verify whether they satisfy the equation or not. If both sides of the equation are equal when the solution is plugged in, then you know the solution is correct. If they are not equal, then there might have been a mistake in the solving process, and you need to go back and review your steps. This practice not only ensures you get the right answer but also strengthens your understanding of the problem-solving process. It's a valuable habit that fosters accuracy and builds confidence in your mathematical abilities. And hey, guys, it’s a really good habit to check your work in every math problem, no just this ones, ok?

Example: Solving (x-5)² = 49

Let's tackle the equation (x−5)2=49(x-5)^2 = 49 using our newfound skills. This example is a classic illustration of how the square root property can simplify complex equations into manageable steps. By following the process carefully, we can unravel the solutions and gain a deeper understanding of the property's application. So, let's put our math hats on and dive into the solution!

  1. The squared term is already isolated! Score! We can skip the first step. This is a great starting point as it allows us to jump straight into the core of the square root property. The expression (x−5)2(x-5)^2 is already by itself on one side of the equation, making our task significantly easier. In many equations, you might need to perform algebraic manipulations to isolate the squared term, but in this case, the setup is already perfect. This saves us time and effort, allowing us to focus directly on the next crucial step: taking the square root of both sides. It's like being handed the perfect tool for the job – everything is in place, and we're ready to proceed with confidence. This efficient start allows us to maintain momentum and tackle the rest of the problem with clarity and precision.
  2. Take the square root of both sides: This gives us (x−5)2=±49\sqrt{(x-5)^2} = \pm\sqrt{49}. Remember that 49\sqrt{49} is both 7 and -7. Applying the square root to both sides is the pivotal moment where we begin to undo the square. By introducing the square root symbol, we're essentially asking: