Solving Radical Equations A Step-by-Step Guide
Hey guys! Today, we're going to break down a fun little math problem together. We've got an equation with a square root in it, and our mission is to find the value of 'x' that makes everything balance out. Don't worry, it's not as scary as it looks! We'll go through each step nice and slow, so you can follow along easily. Math can be super interesting when you approach it the right way, so let's dive in and solve this equation! This is a great exercise in using algebraic principles to isolate variables and understand how different operations affect an equation. By the end of this, you'll not only have the answer, but you'll also have a better grasp of how to tackle similar problems in the future. Let's get started and show that math is something we can all conquer! Remember, the key is to take it one step at a time and stay organized. So, grab your pencils and paper, and let's get to work. We're going to make this equation our friend and discover the secret value hidden within it. Let's do this!
1. Isolate the Square Root: $\sqrt{3x-12} = ?$
Okay, so the first thing we want to do when we see a square root in an equation is to isolate it. That means we want to get the square root part all by itself on one side of the equation. In our case, we have $\sqrt{3x-12} + 17 = 25$. To get the square root alone, we need to get rid of that pesky '+ 17'. How do we do that? We use the magic of inverse operations! Remember, whatever we do to one side of the equation, we have to do to the other to keep things balanced. So, we're going to subtract 17 from both sides. This is a crucial step because it simplifies the equation and brings us closer to solving for 'x'. Think of it like peeling back the layers of an onion – we're gradually uncovering the value of 'x' by undoing the operations that are affecting it. This process of isolating terms is fundamental in algebra, and mastering it will help you tackle a wide range of equations. So, let's subtract 17 from both sides and see what we get. This is where the real fun begins!
When we subtract 17 from both sides of the equation, we get: $\sqrt3x-12} + 17 - 17 = 25 - 17$. The '+ 17' and '- 17' on the left side cancel each other out, leaving us with just the square root. On the right side, 25 minus 17 is 8. So, our equation now looks like this = 8$. Awesome! We've successfully isolated the square root. Now we're one step closer to finding 'x'. This step was all about using our knowledge of inverse operations to simplify the equation. By subtracting 17 from both sides, we've created a cleaner, more manageable equation. This is a common strategy in algebra, and it's super important to understand. You'll use this technique over and over again as you solve more complex problems. So, give yourself a pat on the back – you've just mastered a key skill! Now, let's move on to the next step and see how we can get rid of that square root altogether. We're on a roll!
2. Eliminate the Square Root: $3x - 12 = ?$
Alright, we've got $\sqrt{3x-12} = 8$. Now, how do we get rid of that square root? Well, the inverse operation of taking a square root is squaring! So, if we square both sides of the equation, the square root will disappear. Remember, whatever we do to one side, we have to do to the other to keep things balanced. Squaring both sides is a powerful technique for solving equations with square roots, and it's something you'll use frequently in algebra. Think of it as unlocking a door – the square root is the lock, and squaring is the key. By applying this operation, we're opening up the equation and revealing the expression inside the square root. This step is all about understanding the relationship between square roots and squares, and how they can be used to simplify equations. So, let's square both sides and see what happens. Get ready to watch that square root vanish!
When we square both sides of the equation $\sqrt3x-12} = 8$, we get)^2 = 8^2$. On the left side, the square root and the square cancel each other out, leaving us with just $3x - 12$. On the right side, 8 squared (8 times 8) is 64. So, our equation now looks like this: $3x - 12 = 64$. Fantastic! We've successfully eliminated the square root. Now we have a much simpler equation to solve. This step was crucial because it transformed the equation from one with a square root into a linear equation, which is much easier to handle. This is a common strategy in algebra – to simplify equations by using inverse operations. By squaring both sides, we've made the equation more accessible and brought us closer to finding the value of 'x'. So, give yourself a high-five – you've just mastered another key skill! Now, let's move on to the next step and continue our journey towards solving for 'x'. We're doing great!
3. Isolate the Term with 'x': $3x = ?$
Okay, we're at $3x - 12 = 64$. Our next goal is to isolate the term with 'x', which means we want to get '3x' by itself on one side of the equation. We have that '- 12' hanging around, so we need to get rid of it. Just like before, we'll use inverse operations. The opposite of subtracting 12 is adding 12, so we're going to add 12 to both sides of the equation. Remember, keeping the equation balanced is key! Adding the same value to both sides ensures that the equality remains true. This step is all about strategically using addition to undo subtraction and simplify the equation. Think of it as balancing a scale – we're adding weight to one side to counteract the weight on the other side. This is a fundamental technique in algebra, and it's essential for solving equations. So, let's add 12 to both sides and see what happens. We're on the right track!
When we add 12 to both sides of the equation $3x - 12 = 64$, we get: $3x - 12 + 12 = 64 + 12$. The '- 12' and '+ 12' on the left side cancel each other out, leaving us with just $3x$. On the right side, 64 plus 12 is 76. So, our equation now looks like this: $3x = 76$. Excellent! We've successfully isolated the term with 'x'. We're getting closer and closer to our final answer. This step was about using addition to undo subtraction and simplify the equation. By adding 12 to both sides, we've created a cleaner, more manageable equation. This is a common strategy in algebra, and it's super important to understand. You'll use this technique over and over again as you solve more complex problems. So, give yourself a pat on the back – you've just mastered another key skill! Now, let's move on to the final step and solve for 'x'. We're almost there!
4. Solve for 'x': $x = ?$
We've arrived at $3x = 76$. Now, the final step! To solve for 'x', we need to get 'x' completely by itself. Right now, it's being multiplied by 3. So, what's the inverse operation of multiplication? Division! We're going to divide both sides of the equation by 3 to isolate 'x'. This is the last piece of the puzzle, and it's a crucial step in finding the value of 'x'. Dividing both sides by the same number keeps the equation balanced and allows us to isolate the variable we're solving for. This step is all about understanding the relationship between multiplication and division, and how they can be used to solve equations. Think of it as sharing a pie – we're dividing the pie (76) into 3 equal slices to find the value of one slice ('x'). This is a fundamental technique in algebra, and it's essential for solving equations. So, let's divide both sides by 3 and see what we get. Get ready to find the answer!
When we divide both sides of the equation $3x = 76$ by 3, we get: $\frac3x}{3} = \frac{76}{3}$. On the left side, the 3's cancel each other out, leaving us with just 'x'. On the right side, 76 divided by 3 is $\frac{76}{3}$, which is approximately 25.33. So, our solution is{3}$ or approximately 25.33. We did it! We've successfully solved the equation and found the value of 'x'. This step was about using division to undo multiplication and isolate the variable. By dividing both sides by 3, we've found the exact value of 'x' that makes the equation true. This is a common strategy in algebra, and it's super important to understand. You'll use this technique over and over again as you solve more complex problems. So, give yourself a huge pat on the back – you've just mastered another key skill! We've reached the finish line, and we can proudly say that we've conquered this equation!
Summary of Steps
Let's recap the steps we took to solve the equation $\sqrt{3x-12} + 17 = 25$:
- Isolate the square root: Subtract 17 from both sides to get $\sqrt{3x-12} = 8$.
- Eliminate the square root: Square both sides to get $3x - 12 = 64$.
- Isolate the term with 'x': Add 12 to both sides to get $3x = 76$.
- Solve for 'x': Divide both sides by 3 to get $x = \frac{76}{3}$.
So, the solution to the equation is $x = \frac{76}{3}$, or approximately 25.33.
Discussion Category: Mathematics
This problem falls under the category of mathematics, specifically algebra. It involves solving an equation with a square root, which requires understanding inverse operations and algebraic manipulation. These skills are fundamental to many areas of mathematics and are essential for further study in STEM fields.
