Solving Equations Square Root Of Both Sides Of (x+9)^2=25

Hey guys! Let's dive into a common algebra problem that many students encounter: solving equations by taking the square root. Today, we’re going to break down the equation $(x+9)^2=25$ and figure out which option is the result of taking the square root of both sides. This is a fundamental concept in algebra, and understanding it thoroughly will help you tackle more complex problems down the road. So, let’s get started and make sure we nail this! Solving equations often involves reversing operations, and taking the square root is the inverse of squaring a quantity. But there’s a little trick to it, which we’ll uncover step by step.

Understanding the Square Root Property

Before we jump into the specific problem, let’s quickly review the square root property. This property is the key to understanding how to solve equations like the one we have. The square root property states that if you have an equation in the form $a^2 = b$, then $a = \_\pm \sqrt{b}$. Notice the $\_\pm$ symbol? This is super important! When we take the square root of a number, we need to consider both the positive and negative roots. For example, the square root of 25 is both 5 and -5 because $5^2 = 25$ and $(-5)^2 = 25$. This is because squaring a negative number results in a positive number. Ignoring the negative root is a common mistake, so always remember to include both possibilities. This might seem like a small detail, but it can completely change the solution set of an equation. Understanding this concept deeply is crucial not only for solving quadratic equations but also for many other areas of mathematics, including calculus and complex numbers. So, let’s make sure we’ve got this down pat before moving forward!

Why Both Positive and Negative Roots Matter

To really grasp this, think about it this way: When you square a number, you’re essentially multiplying it by itself. Whether that number is positive or negative, the result will be positive. Therefore, when you reverse the process by taking the square root, you have to account for both scenarios. If we only considered the positive root, we'd be missing half of the potential solutions. This is why the $\_\pm$ symbol is so critical. It's a reminder that there are usually two numbers that, when squared, give you the same positive result. Imagine if we were solving a real-world problem, like finding the dimensions of a square with a certain area. There might be two possible side lengths (one positive and one negative), although in the physical world, we'd typically only consider the positive one. However, in the abstract world of algebra, both solutions are valid and important. Keeping this in mind will save you from making errors and ensure you find all possible solutions to your equations. So, always remember the dual nature of square roots!

Solving $(x+9)^2=25$ Step-by-Step

Now, let’s apply this knowledge to our equation: $(x+9)^2=25$. The first step is to take the square root of both sides of the equation. Remember, whatever you do to one side, you must do to the other to maintain the balance. So, we get:

$\sqrt{(x+9)^2} = \pm \sqrt{25}$

On the left side, the square root cancels out the square, leaving us with $(x+9)$. On the right side, the square root of 25 is 5, but we need to remember both the positive and negative roots, so we have $\_\pm 5$. This gives us:

$x+9 = \pm 5$

And just like that, we’ve arrived at the equation that results from taking the square root of both sides of our original equation! Notice how crucial it was to include the $\_\pm$ sign. Without it, we’d only have half the picture. This step-by-step approach helps to break down the problem into manageable parts, making it easier to understand and solve. It's a good practice to always show your work, as it allows you to track your progress and identify any potential errors. Plus, it helps in reinforcing the concepts in your mind. So, let's keep moving and see how this fits into our multiple-choice options.

Comparing Our Result with the Given Options

Looking back at the options provided in the original question:

A. $x+3= \pm 5$ B. $x+3= \pm 25$ C. $x+9= \pm 5$ D. $x+9= \pm 25$

We can clearly see that our result, $x+9 = \pm 5$, matches option C. Options A and B have an incorrect left-hand side ($x+3$ instead of $x+9$), and option D has an incorrect right-hand side ($\_\pm 25$ instead of $\_\pm 5$). This highlights the importance of careful calculation and attention to detail when solving equations. Even a small error can lead to a completely different answer. By comparing our solution step-by-step with the options, we can confidently identify the correct one. This process not only gives us the answer but also helps us verify our work and ensure accuracy. So, always take the time to compare your results with the available options – it’s a great way to double-check your work!

Solving for x: The Next Step

Now that we've found the correct equation, let's take it a step further and actually solve for $x$. This will give us a complete solution and reinforce the concepts we've discussed. We have the equation:

$x+9 = \pm 5$

This equation actually represents two separate equations:

1. $x + 9 = 5$
2. $x + 9 = -5$

To solve each equation, we need to isolate $x$ by subtracting 9 from both sides. Let's start with the first equation:

$x + 9 - 9 = 5 - 9$ $x = -4$

Now, let's solve the second equation:

$x + 9 - 9 = -5 - 9$ $x = -14$

So, we have two solutions for $x$: $x = -4$ and $x = -14$. These are the values that, when plugged back into the original equation, will make it true. Solving for $x$ not only gives us the final answer but also demonstrates the power and elegance of algebraic manipulation. It's like piecing together a puzzle – each step brings us closer to the complete picture. By understanding how to isolate variables, we can unlock the solutions to a wide range of problems. So, let's keep practicing and building our skills!

Verifying the Solutions

To be absolutely sure we've got the correct solutions, it’s always a good idea to plug them back into the original equation and check. This process is called verification, and it's a crucial step in problem-solving. It helps us catch any errors we might have made along the way. Let's start with our first solution, $x = -4$:

$(-4 + 9)^2 = 25$ $(5)^2 = 25$ $25 = 25$

This checks out! Now, let's verify our second solution, $x = -14$:

$(-14 + 9)^2 = 25$ $(-5)^2 = 25$ $25 = 25$

This one also checks out! Both solutions satisfy the original equation, which means we've solved it correctly. Verification is like the final seal of approval on our work. It gives us confidence that our answers are accurate and that we've understood the problem fully. It's a habit that every successful problem-solver should cultivate. So, always remember to verify your solutions – it's the best way to avoid mistakes and ensure success!

Common Mistakes to Avoid

When solving equations by taking square roots, there are a few common pitfalls that students often encounter. Being aware of these mistakes can help you avoid them and improve your problem-solving skills. One of the most frequent errors is forgetting to include the $\_\pm$ sign when taking the square root. As we’ve discussed, this means missing half of the solutions. Always remember that the square root of a positive number has both a positive and a negative value. Another common mistake is incorrectly applying the square root property to more complex expressions. For example, you can't simply take the square root of each term in an expression like $(x+9)^2$ separately. You need to treat the entire expression inside the parentheses as a single unit. Similarly, students sometimes make errors when simplifying square roots or dealing with negative numbers under the square root. A solid understanding of basic algebraic principles and careful attention to detail are essential to avoid these mistakes. By being mindful of these common pitfalls, you can improve your accuracy and confidence in solving equations.

Tips for Success

To master solving equations by taking square roots, here are a few tips that can help:

1. Always remember the $\_\pm$ sign: This is the most crucial step, so make it a habit.
2. Isolate the squared term: Before taking the square root, make sure the squared expression is by itself on one side of the equation.
3. Show your work: Writing down each step helps you keep track of your progress and identify errors.
4. Verify your solutions: Plug your answers back into the original equation to check if they are correct.
5. Practice regularly: The more you practice, the more comfortable you'll become with the process.

By following these tips, you can develop a systematic approach to solving equations and build your confidence in algebra. Remember, mathematics is like learning a new language – it takes time and practice to become fluent. So, don't get discouraged if you make mistakes along the way. Instead, learn from them and keep practicing. With dedication and perseverance, you'll be able to tackle even the most challenging problems!

Conclusion

So, to answer the original question, the equation that results from taking the square root of both sides of $(x+9)^2=25$ is C. $x+9= \pm 5$. We’ve not only identified the correct answer but also walked through the entire process, from understanding the square root property to verifying our solutions. This comprehensive approach ensures that you not only know the answer but also understand the underlying concepts. Remember, the key to success in algebra is understanding the fundamentals and practicing regularly. Keep these principles in mind, and you'll be well-equipped to tackle any equation that comes your way. Happy solving, guys! And always remember, every problem is an opportunity to learn and grow.