Solving (x-10)^2=-9 By The Square Root Property

Hey guys! Today, we're diving deep into the world of algebra to tackle a specific type of equation: those we can solve using the square root property. This method is super handy when you have a squared term isolated on one side of the equation. We'll break down the concept, walk through the steps, and then apply it to a practice problem. So, grab your calculators (or your brainpower!) and let's get started!

The square root property is a powerful tool in algebra that allows us to solve equations where a variable expression is squared. This method really shines when you've got something like $(x + a)^2 = b$, where 'a' and 'b' are constants. The core idea is that if two things are equal, their square roots must also be equal (or opposites of each other, because squaring a negative number also results in a positive one). This is where the $\pm$ symbol comes in, indicating both the positive and negative roots.

To effectively wield the square root property, there's a sequence of steps you'll want to follow. First, and this is crucial, you need to isolate the squared term. This means getting the expression that's being squared all by itself on one side of the equation. Think of it like getting the VIP section ready before the party starts – no distractions allowed! This might involve adding, subtracting, multiplying, or dividing terms to both sides of the equation.

Once the squared term is isolated, the magic begins: take the square root of both sides of the equation. Remember that when you take the square root, you need to consider both the positive and negative possibilities, hence the $\pm$ symbol. This is the heart of the square root property. For example, if you have $x^2 = 9$, taking the square root of both sides gives you $x = \pm 3$, because both 3 squared and -3 squared equal 9. Don't forget this crucial step, or you'll miss half of your solutions!

After taking the square root, you'll likely have some simplification to do. This might involve simplifying radicals (like turning $\sqrt{8}$ into $2\sqrt{2}$) or dealing with imaginary numbers if you encounter the square root of a negative number. Remember, the square root of -1 is denoted by the imaginary unit 'i'. So, $\sqrt{-9}$ becomes 3i. Finally, you'll need to solve for the variable. This might involve adding, subtracting, multiplying, or dividing to get 'x' all by itself. You'll often end up with two solutions, one from the positive square root and one from the negative square root.

Understanding the nuances of the square root property opens up a whole new world of algebraic problem-solving. It's not just about memorizing steps; it's about understanding the underlying logic. Why do we isolate the squared term? Because we need to undo the squaring operation with its inverse, the square root. Why do we consider both positive and negative roots? Because both a positive and a negative number, when squared, result in a positive number. Grasping these concepts will make you a much more confident and capable equation solver. And remember, practice makes perfect! The more you use the square root property, the more comfortable and proficient you'll become.

Applying the Square Root Property to a Specific Equation

Alright, let's put our newfound knowledge into action! We're going to tackle the equation $(x - 10)^2 = -9$. This problem is perfect for demonstrating the square root property because we have a squared term, $(x - 10)^2$, nicely isolated on the left side of the equation. Our mission is to find the values of 'x' that make this equation true. So, let's break it down step by step.

Remember the first crucial step: isolate the squared term. But hey, good news! In this case, $(x - 10)^2$ is already by its lonesome on the left side. This means we can jump straight to the exciting part – taking the square root! This is where the square root property really shines. We're going to apply the golden rule of algebra: what you do to one side, you must do to the other. So, we'll take the square root of both sides of the equation.

Taking the square root of $(x - 10)^2$ is pretty straightforward. The square root and the square undo each other, leaving us with $(x - 10)$ on the left side. But on the right side, we have the square root of -9. Uh oh, a negative number under the square root? Don't panic! This is where imaginary numbers come into play. Remember that $\sqrt{-1}$ is defined as the imaginary unit 'i'. So, $\sqrt{-9}$ can be rewritten as $\sqrt{9 * -1}$, which is the same as $\sqrt{9} * \sqrt{-1}$, which simplifies to 3i. And don't forget the crucial $\pm$ symbol! We need to account for both the positive and negative square roots, so we have $\pm 3i$.

Now our equation looks like this: $x - 10 = \pm 3i$. We're almost there! The final step is to solve for x. To get 'x' by itself, we simply need to add 10 to both sides of the equation. This gives us $x = 10 \pm 3i$. And there you have it! We've successfully solved the equation using the square root property.

But wait, what does $x = 10 \pm 3i$ actually mean? It means we have two solutions: $x = 10 + 3i$ and $x = 10 - 3i$. These are complex numbers, consisting of a real part (10) and an imaginary part (3i or -3i). Complex solutions are a common occurrence when dealing with square roots of negative numbers. They might seem a bit strange at first, but they're perfectly valid solutions in the world of algebra!

Let's recap the whole process. We started with the equation $(x - 10)^2 = -9$. We recognized that the squared term was already isolated. We took the square root of both sides, remembering the crucial $\pm$ sign. We dealt with the square root of -9 by introducing the imaginary unit 'i'. And finally, we solved for 'x' by adding 10 to both sides, resulting in the complex solutions $x = 10 + 3i$ and $x = 10 - 3i$. This journey beautifully illustrates the power and elegance of the square root property.

Identifying the Correct Solution from the Given Options

Now that we've meticulously solved the equation $(x - 10)^2 = -9$, let's match our solution with the options provided. This is a crucial step in any problem-solving process – ensuring that your hard work translates into selecting the correct answer. We've already determined that the solutions are $x = 10 + 3i$ and $x = 10 - 3i$, which can be compactly written as $x = 10 \pm 3i$.

Looking at the options, we have:

• A. $\{-10 \pm 3i\}$
• B. $\{\pm \frac{3i}{10}\}$
• C. $\{10i \pm 3\}$
• D. $\{10 \pm 3i\}$

By a simple comparison, it's clear that option D, $\{10 \pm 3i\}$, perfectly matches our solution. The other options are incorrect. Option A has the wrong real part (-10 instead of 10). Option B is completely different, involving division and a different structure. Option C mixes up the real and imaginary parts and includes 'i' in the wrong place.

This exercise highlights the importance of not just solving the equation but also carefully comparing your result with the given options. It's easy to make a small mistake in the final step, so double-checking is always a good idea. Think of it as the final polish on a masterpiece – ensuring that every detail is just right!

Choosing the correct option reinforces our understanding of the square root property and how it leads to accurate solutions. It's a testament to the power of step-by-step problem-solving and the importance of meticulousness in mathematics. So, pat yourself on the back – you've successfully navigated this algebraic challenge!

Key Takeaways and Further Practice

We've journeyed through the process of solving equations using the square root property, and hopefully, you've gained a solid understanding of this valuable technique. Before we wrap up, let's highlight the key takeaways and discuss how you can further hone your skills. Remember, practice is the secret sauce to mastering any mathematical concept!

The most crucial takeaway is the square root property itself: if you have an equation in the form $(expression)^2 = constant$, you can solve for the variable by taking the square root of both sides, remembering the all-important $\pm$ symbol. This property is a direct consequence of the relationship between squaring and square roots – they're inverse operations that undo each other.

Isolating the squared term is another critical step. You can't apply the square root property until the squared expression is all by itself on one side of the equation. Think of it as laying the foundation before building a house – it's a necessary prerequisite. This often involves algebraic manipulations like adding, subtracting, multiplying, or dividing terms to both sides of the equation.

Don't forget the $\pm$! This little symbol is a powerhouse, reminding us that both positive and negative numbers, when squared, result in a positive number. Missing the $\pm$ means missing half of your solutions, which is definitely not what we want. It's like only painting half of a picture – incomplete and unsatisfying.

Be prepared to encounter imaginary numbers. When you take the square root of a negative number, you'll venture into the realm of complex numbers, involving the imaginary unit 'i'. Embrace these numbers! They're a fascinating and essential part of the mathematical landscape. Think of them as adding a splash of vibrant color to your mathematical palette.

To solidify your understanding, the best thing you can do is practice, practice, practice! Seek out more equations that can be solved using the square root property. Start with simpler examples and gradually work your way up to more complex ones. Look for patterns and connections. The more you practice, the more intuitive this method will become.

Consider exploring variations of the square root property, such as solving equations involving cube roots or other radicals. The underlying principle remains the same: using inverse operations to isolate the variable. This broader perspective will deepen your understanding of algebraic problem-solving as a whole. You can also try making up your own problems and solving them. This is a great way to test your understanding and identify areas where you might need more practice.

So, keep practicing, keep exploring, and keep embracing the beauty and power of mathematics! The square root property is just one tool in your ever-expanding algebraic toolbox. With dedication and perseverance, you'll become a master equation solver in no time! Remember, every problem you solve is a step forward on your mathematical journey.