Solving For X: X² - 6 = 30

Hey guys! Let's dive into this math problem together. We're going to figure out how to solve for x when we're given the equation x² - 6 = 30. It might seem a little tricky at first, but trust me, it's totally manageable and we'll break it down step-by-step. We aim to make this process as clear and easy as possible, so you can tackle similar problems with confidence. Whether you're a student brushing up on your algebra skills or just someone who enjoys a good math challenge, you're in the right place. We'll go through each step meticulously, ensuring you understand not just how to solve it, but also why each step is necessary. So, grab your pencils, and let's get started!

Understanding the Equation

Okay, so the first thing we see is the equation x² - 6 = 30. This is a quadratic equation, meaning it involves a variable (in this case, x) raised to the power of 2. The goal here is to isolate x, which means getting x by itself on one side of the equation. To do this, we need to undo the operations that are being applied to x. Remember, whatever we do to one side of the equation, we have to do to the other side to keep things balanced. Think of it like a scale – if you add weight to one side, you need to add the same weight to the other side to keep it even. In this equation, we have x squared, and then we're subtracting 6. Our first step will be to get rid of that subtraction. Understanding the structure of the equation is key to knowing how to approach it. By recognizing it as a quadratic equation, we know we might end up with two possible solutions for x, since squaring a number can result in the same value whether the number is positive or negative. Keep this in mind as we move forward, and let's get to the next step!

Step 1: Isolating the x² Term

Alright, let's isolate the x² term. Currently, we have x² - 6 = 30. To get x² by itself, we need to get rid of that -6. How do we do that? By adding 6 to both sides of the equation! Remember, whatever we do to one side, we have to do to the other. So, we add 6 to both sides like this:

x² - 6 + 6 = 30 + 6

This simplifies to:

x² = 36

Awesome! Now we have x² all by itself on one side of the equation. This is a huge step because it brings us closer to solving for x. By adding 6 to both sides, we've effectively canceled out the -6 on the left side, leaving us with just x². This makes the next step much easier. Now that we have x² = 36, we need to figure out what number, when multiplied by itself, equals 36. This is where we'll use the inverse operation of squaring, which is taking the square root. Keep following along, and we'll get there in the next step!

Step 2: Taking the Square Root

Okay, now we've got x² = 36. To find x, we need to take the square root of both sides of the equation. Remember, the square root of a number is a value that, when multiplied by itself, gives you the original number. The square root of x² is simply x. But here's the crucial part: when we take the square root of a number, we have to consider both the positive and negative roots. Why? Because both a positive number and its negative counterpart, when squared, will give you a positive result. For example, both 6 * 6 and -6 * -6 equal 36.

So, when we take the square root of both sides, we get:

x = ±√36

This means x could be either the positive square root of 36 or the negative square root of 36. And since the square root of 36 is 6, we have:

x = ±6

This tells us that x = 6 or x = -6. Always remember to consider both positive and negative roots when solving equations like this; otherwise, you might miss a valid solution!

Step 3: The Solutions

So, after taking the square root, we found that x = ±6. This means we have two possible solutions for x: x = 6 and x = -6. Let's break down why both of these are valid. If we substitute x = 6 back into the original equation, we get:

(6)² - 6 = 36 - 6 = 30

Yep, that checks out! Now let's try x = -6:

(-6)² - 6 = 36 - 6 = 30

That also works! Both values satisfy the original equation, which means both are correct solutions. When you're solving quadratic equations, it's common to have two solutions because squaring a number always results in a positive value, regardless of whether the original number was positive or negative. Therefore, both 6 and -6 are valid solutions for x. So, the final answer is that x can be either 6 or -6.

Final Answer

Alright, guys, after walking through each step, we've nailed it! The solutions to the equation x² - 6 = 30 are x = 6 and x = -6. Remember, when you're dealing with quadratic equations like this, always consider both the positive and negative square roots. This is super important to ensure you don't miss any possible solutions. We started by isolating the x² term by adding 6 to both sides of the equation. Then, we took the square root of both sides, remembering to account for both positive and negative roots. Finally, we verified our solutions by plugging them back into the original equation to make sure they worked. Solving for x can sometimes feel like a puzzle, but with a bit of practice and a clear understanding of the steps involved, you'll be solving these equations like a pro in no time. Keep practicing, and you'll become more confident with each problem you solve. Great job, everyone! You've successfully solved for x!