Solving -3b² + 25 When B = 7: A Step-by-Step Guide
Hey there, math enthusiasts! Today, we're diving into a classic algebra problem. We'll find the value of the expression -3b² + 25 when b equals 7. It's a fundamental concept, but let's make sure we break it down in a way that's super clear and easy to follow. Remember, understanding the order of operations is key here. It's like following a recipe – you need to do things in the right order to get the perfect result. So, grab your pencils (or your favorite digital notepad!), and let's get started. We'll explore the problem step by step, ensuring you grasp not just the answer but also the how and why behind it.
Understanding the Problem: The Core Concepts
Let's start by making sure we're all on the same page. The expression -3b² + 25 involves several mathematical operations: multiplication, exponentiation (that's the little '2' up in b²), and addition. When we're given that b = 7, we're being asked to substitute 7 every time we see b in the expression and then simplify. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), tells us the sequence in which we must perform these operations. This order is crucial because changing it will change the answer! Exponents are evaluated before multiplication and, subsequently, before addition. This problem is a beautiful example of how simple rules and careful execution lead to an accurate solution. The value of b is a specific number, and the task at hand is to substitute this number into our expression and find its value. This is how we tackle this problem and ensure the solution is correct.
Now, let's look at the options. We have four options to choose from, A) 172, B) 122, C) 17, and D) -122. We will find out which one of these options is correct by substituting the given values into the expression and evaluating it. The goal is to accurately calculate the value of the expression -3b² + 25 when b = 7. It seems simple, right? It really is, once you break it down into steps. Let's start with the first step.
Step-by-Step Solution: Breaking Down the Math
Here's how we'll solve this problem, step by step, to ensure we get to the correct answer. The key is to be methodical and careful. We start by substituting the value of b (which is 7) into the expression -3b² + 25. So, instead of b, we'll write 7: -3(7)² + 25. Now, let's look at each part of the expression more closely.
- Exponents First: Following the order of operations (PEMDAS), we start with the exponent. We need to calculate 7². This means 7 multiplied by itself: 7 * 7 = 49. So, our expression now becomes -3(49) + 25.
- Multiplication: Next, we tackle the multiplication. We need to multiply -3 by 49: -3 * 49 = -147. Our expression simplifies to -147 + 25.
- Addition: Finally, we perform the addition. We add -147 and 25: -147 + 25 = -122.
And there we have it! The value of the expression -3b² + 25 when b = 7 is -122. See? It's all about taking it one step at a time and remembering your rules. Each step gets us closer to the final solution. The order of these operations is crucial for getting the right answer. Incorrect sequencing will provide an incorrect result. It's like a chain – each link must be in the right place to be strong. Once these three steps are done, the answer is easily found. The solution to the problem, the expression when b = 7, is -122.
Examining the Answer Choices
Now that we've found our answer, let's look back at the options provided to see which one matches our result. Remember, we calculated that the value of the expression -3b² + 25 when b = 7 is -122.
- Option A) 172: This is incorrect. It's a completely different number than what we calculated.
- Option B) 122: This is also incorrect. It's the positive version of the answer, but the negative sign is important in our solution.
- Option C) 17: This is incorrect as well. This number comes from doing some of the operations in the wrong order.
- Option D) -122: Bingo! This matches our calculation exactly. This option correctly reflects the accurate value of the expression when b = 7.
Therefore, by working through the steps and following the order of operations, we've successfully matched our calculated answer with one of the provided options. It's always a good practice to revisit the options and confirm your solution. This will make sure you have chosen the best answer, and that it reflects the answer you calculated when working out the problem.
Conclusion: Mastering the Expression
So, there you have it! We've successfully navigated through the expression -3b² + 25 when b = 7, breaking down each step to arrive at the solution. The correct answer is D) -122. We've learned that understanding the order of operations is key, and that taking things step by step makes complex-looking problems much more manageable. Remember, math is like a language; with practice, you'll become fluent. Keep practicing and applying these concepts to new problems. The more you work with these types of expressions, the more comfortable and confident you'll become. Each time you solve a problem, you're building a stronger foundation in algebra. Keep practicing the problems and you'll find that your skills will keep improving. Understanding the solution to this problem gives you a solid foundation and you will be able to do this again and again with no problems.
We hope this guide has been helpful. Keep up the great work, and happy calculating!
For further exploration of algebraic expressions and the order of operations, you may find the following resource beneficial:
