Solving Equations: X² – 0.1 = 0.06 And (x – 1)² = 36

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Solving Equations: x² – 0.1 = 0.06 and (x – 1)² = 36

Hey guys! Today, we're diving into the world of algebra to solve some equations. Specifically, we're going to tackle two equations: a) x² – 0.1 = 0.06 and b) (x – 1)² = 36. Don't worry if these look a bit intimidating at first. We'll break them down step by step, so you'll be solving quadratic equations like a pro in no time! Understanding how to solve these types of equations is super important because they pop up everywhere in math and even in real-world applications. Think about physics, engineering, or even economics – quadratic equations are the unsung heroes behind many calculations. So, let’s roll up our sleeves and get started! We’ll explore the fundamental concepts, the step-by-step solutions, and the reasoning behind each move we make. By the end of this, you’ll not only be able to solve these specific equations but also have a solid foundation for tackling similar problems in the future. Remember, practice makes perfect, so don't be afraid to grab a pencil and paper and work along with us. The key is to understand the process, not just memorize the answers. So, let's get to it and make math a little less mysterious and a lot more fun!

Part A: Solving x² – 0.1 = 0.06

Okay, let's start with the first equation: x² – 0.1 = 0.06. The goal here is to isolate x, right? We want to get x all by itself on one side of the equation. To do that, we need to undo the operations that are affecting x. In this case, we have a subtraction and a square. Remember, we always tackle things in reverse order of operations (PEMDAS/BODMAS), so we'll deal with the subtraction first.

Step 1: Isolate x²

To isolate x², we need to get rid of that -0.1. The opposite of subtracting 0.1 is adding 0.1, so we'll add 0.1 to both sides of the equation. This is a crucial step because whatever we do to one side of the equation, we must do to the other to keep things balanced. Think of it like a seesaw – if you add weight to one side, you need to add the same weight to the other side to keep it level. So, our equation now looks like this:

x² – 0.1 + 0.1 = 0.06 + 0.1

Simplifying both sides, we get:

x² = 0.16

Great! We've successfully isolated x². Now, we're one step closer to finding x. But we're not quite there yet. We need to get rid of that square. That's where the next step comes in.

Step 2: Take the Square Root

The opposite of squaring a number is taking its square root. So, to get x by itself, we need to take the square root of both sides of the equation. Now, this is a bit tricky because every positive number has two square roots – a positive one and a negative one. For example, both 4 and -4, when squared, give you 16. So, we need to remember to consider both possibilities when we take the square root. Applying the square root to both sides, we get:

√(x²) = ±√(0.16)

This simplifies to:

x = ±0.4

And there you have it! We've found our solutions for x. It can be either 0.4 or -0.4. These are the two numbers that, when squared, will give you 0.16. Remember to always consider both positive and negative roots when solving equations like this. It's a common mistake to forget the negative root, so keep that in mind. Now, let’s move on to the second equation and see how we can tackle that one.

Part B: Solving (x – 1)² = 36

Alright, let's move on to the second equation: (x – 1)² = 36. This one looks a little different from the first, but the same principles apply. Our goal is still to isolate x, but this time, we have a squared term and something being subtracted inside the parentheses. There are a couple of ways we can approach this, but we'll go with the method that's often the most straightforward: taking the square root first.

Step 1: Take the Square Root

Just like before, we want to undo the squaring operation. So, we'll take the square root of both sides of the equation. And remember, we need to consider both the positive and negative square roots. So, we have:

√((x – 1)²) = ±√36

This simplifies to:

(x – 1) = ±6

Now, we're left with a much simpler equation. We've gotten rid of the square, but we still need to isolate x. We have two possibilities to consider here: x – 1 = 6 and x – 1 = -6. Let's tackle each of these separately.

Step 2a: Solve for x when (x – 1) = 6

To solve x – 1 = 6, we need to get rid of that -1. The opposite of subtracting 1 is adding 1, so we'll add 1 to both sides of the equation:

x – 1 + 1 = 6 + 1

This simplifies to:

x = 7

So, one solution for x is 7. Now, let's look at the other possibility.

Step 2b: Solve for x when (x – 1) = -6

To solve x – 1 = -6, we do the same thing – add 1 to both sides of the equation:

x – 1 + 1 = -6 + 1

This simplifies to:

x = -5

So, our other solution for x is -5. We've found both solutions for this equation! Remember, when you take the square root of both sides, you often end up with two possible solutions, so it's important to consider both the positive and negative cases. Now, let's recap what we've done and talk about why these steps work.

Summary and Key Concepts

Okay, guys, let’s take a step back and recap what we’ve done. We successfully solved two equations today: x² – 0.1 = 0.06 and (x – 1)² = 36. For the first equation, we isolated x² by adding 0.1 to both sides and then took the square root, remembering to consider both positive and negative roots. For the second equation, we started by taking the square root of both sides, which gave us two separate equations to solve. We then isolated x in each case to find our two solutions.

Key Concepts Recap

  • Isolating the Variable: The main goal in solving any equation is to get the variable (in this case, x) all by itself on one side of the equation. We do this by undoing the operations that are affecting the variable, one step at a time.
  • Inverse Operations: We use inverse operations to undo operations. For example, addition undoes subtraction, multiplication undoes division, and taking the square root undoes squaring. It’s like having a mathematical toolkit where each tool is designed to reverse another.
  • Maintaining Balance: Whatever you do to one side of an equation, you must do to the other side to keep the equation balanced. This is a fundamental principle in algebra and is crucial for solving equations correctly.
  • Square Roots: Positive and Negative: When taking the square root of a number, remember to consider both the positive and negative roots. This is because both the positive and negative versions of a number, when squared, will give you the same positive result. Failing to consider both roots is a common mistake, so always keep it in mind.

Why These Steps Work

The reason these steps work is rooted in the basic properties of equality. The properties of equality allow us to perform the same operation on both sides of an equation without changing the solution. This ensures that we're always working with an equivalent equation, just in a simpler form. For example, when we add the same number to both sides of an equation, we're using the addition property of equality. When we take the square root of both sides, we're applying a property that says if two things are equal, then their square roots must also be equal (or the negatives of their square roots).

Practice Makes Perfect

Now that we've walked through these examples, the best way to solidify your understanding is to practice! Grab some similar equations and try solving them on your own. Don't be afraid to make mistakes – that's how we learn. And if you get stuck, go back and review the steps we took in these examples. Remember, solving equations is a skill that gets better with practice. The more you do it, the more comfortable you'll become with the process. And the more comfortable you are, the more confident you'll feel tackling more complex problems. Think of it like learning a new language or a musical instrument – it takes time and effort, but the rewards are well worth it.

Where to Find Practice Problems

  • Textbooks: Your algebra textbook is a great resource for practice problems. Look for sections on solving equations and try some of the examples.
  • Online Resources: There are tons of websites and apps that offer practice problems and step-by-step solutions. Khan Academy is a fantastic resource for math topics of all kinds.
  • Worksheets: Search online for algebra worksheets. You can find worksheets with a variety of problems to challenge yourself.

Conclusion

So, there you have it, guys! We've successfully solved two quadratic equations and recapped some key algebraic concepts. Remember, solving equations is a fundamental skill in math, and it's something you'll use again and again. By understanding the principles behind the steps we took, you'll be well-equipped to tackle a wide range of algebraic problems. Keep practicing, stay curious, and don't be afraid to ask for help when you need it. Math can be challenging, but it's also incredibly rewarding. And with a little effort and perseverance, you can conquer any equation that comes your way. Keep up the great work, and I'll catch you in the next algebra adventure!