Solving For B: How To Solve √2b = 3
Hey guys! Let's break down how to solve the equation √2b = 3. This is a common type of algebra problem, and understanding the steps will help you tackle similar questions with confidence. We'll go through each step in detail, so you can follow along easily. Grab your pencils and let's get started!
Understanding the Problem
Before we jump into solving, let's make sure we understand what the problem is asking. We have the equation √2b = 3, and our goal is to find the value of 'b' that makes this equation true. In other words, we need to isolate 'b' on one side of the equation. The main challenge here is dealing with the square root. Remember, the square root of a number is a value that, when multiplied by itself, gives you the original number. For example, the square root of 9 is 3 because 3 * 3 = 9.
Keywords are your friends here! When you see 'solve for,' it means you need to isolate the variable. When you see a square root, you'll likely need to square both sides of the equation at some point. Keeping these keywords in mind will guide you through the process. So, to reiterate, we are given the equation √2b = 3, and we need to find the value of b that satisfies this equation. The variable b is currently under a square root, which means our initial steps will involve getting rid of that square root to isolate b. Let's dive into the actual steps now and see how we can accomplish this.
Step-by-Step Solution
Okay, let’s get to the fun part – solving for b. Here’s how we do it, step-by-step:
Step 1: Square Both Sides
To get rid of the square root, we need to do the opposite operation, which is squaring. So, we square both sides of the equation. This means we raise both the left side (√2b) and the right side (3) to the power of 2.
(√2b)² = 3²
When you square a square root, they cancel each other out. So, (√2b)² simplifies to 2b. And 3² is simply 3 * 3, which equals 9.
So, our equation now looks like this:
2b = 9
Squaring both sides is a crucial step. By squaring both sides of the equation √2b = 3, we eliminate the square root, which is essential for isolating the variable b. This step simplifies the equation to 2b = 9, making it much easier to solve for b. Always remember that whatever operation you perform on one side of the equation, you must perform the same operation on the other side to maintain the equality. This principle ensures that the solution remains valid. In this case, squaring both sides keeps the equation balanced and allows us to proceed toward finding the value of b.
Step 2: Isolate b
Now, we want to get b all by itself on one side of the equation. Currently, b is being multiplied by 2. To undo this multiplication, we need to divide both sides of the equation by 2.
2b / 2 = 9 / 2
On the left side, the 2s cancel each other out, leaving us with just b. On the right side, 9 divided by 2 is 4.5.
So, we have:
b = 9 / 2 = 4.5
Isolating b involves performing the inverse operation to what's currently affecting b. Since b is being multiplied by 2, we divide both sides by 2. This division isolates b on one side of the equation, giving us b = 9/2. Simplifying this fraction, we find that b = 4.5. This result is the solution to the original equation. The goal of solving any equation is to isolate the variable, and in this case, we've successfully found the value of b that makes the equation true. Remember, always check your work by substituting the value back into the original equation to ensure it holds.
Step 3: Check Your Answer
It’s always a good idea to check your answer to make sure it’s correct. To do this, we plug our value for b (which is 4.5) back into the original equation:
√2b = 3
√2 * 4.5 = 3
√9 = 3
3 = 3
Since the equation holds true, our answer is correct!
Checking the answer is a crucial step in solving equations. By substituting the value of b we found (which is 4.5) back into the original equation √2b = 3, we can verify whether our solution is correct. When we substitute 4.5 for b, we get √2 * 4.5 = √9, which simplifies to 3 = 3. Since both sides of the equation are equal, this confirms that our solution b = 4.5 is indeed correct. Always take the time to check your work, as it helps prevent errors and ensures accuracy in your solutions. This practice builds confidence and reinforces your understanding of the problem-solving process.
Final Answer
So, the final answer is:
b = 4.5
Practice Problems
Want to test your skills? Try solving these similar equations:
- √3x = 6
- √5y = 10
- √4z = 8
Solving these practice problems will solidify your understanding of how to solve equations with square roots. Each problem requires you to isolate the variable by squaring both sides of the equation and then performing the necessary operations to solve for the variable. Remember to check your answers by substituting the values back into the original equations. Practice makes perfect, and working through these problems will help you become more confident in your ability to tackle similar algebraic challenges. Additionally, understanding these concepts builds a strong foundation for more advanced topics in mathematics. So, grab a pen and paper, and let's get practicing!
Conclusion
Solving for b in the equation √2b = 3 involves a few simple steps: squaring both sides to get rid of the square root, isolating b, and then checking your answer. Once you understand these steps, you can solve similar problems with ease. Keep practicing, and you'll become a pro at algebra in no time!