Solving (x + 4)² + X² + 16 When X = 2 A Step-by-Step Guide

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Hey guys! Today, we're diving into a fun math problem that involves evaluating an algebraic expression. Let's break it down step by step so you can see exactly how we arrive at the solution. We're going to tackle the expression (x + 4)² + x² + 16, and our mission is to find out what it equals when x = 2. Sounds like a plan? Let's get started!

Understanding the Problem

Before we jump into the calculations, let's make sure we fully grasp what the problem is asking. We have an algebraic expression, which is a combination of variables (in this case, just x), numbers, and mathematical operations (addition, subtraction, multiplication, exponentiation, etc.). Our job is to substitute the value 2 for every instance of x in the expression and then simplify it using the correct order of operations. This will give us a numerical answer.

The expression we're working with is (x + 4)² + x² + 16. Notice the parentheses, the exponent, and the addition signs. These are all clues that tell us how to proceed. Remember, the order of operations (often remembered by the acronym PEMDAS/BODMAS) is crucial: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

The question provides us with multiple-choice options: A) 36, B) 40, C) 44, and D) 48. One of these is the correct answer, and our step-by-step solution will reveal which one it is. So, let's roll up our sleeves and get calculating!

Step-by-Step Solution

Okay, let's dive into the nitty-gritty and solve this expression step-by-step. This is where we put our math hats on and get our hands dirty with the calculations. Remember, the key to solving any math problem is to take it one step at a time, so we don't make any silly mistakes. Let's break it down:

  1. Substitution: The very first thing we need to do is replace every x in the expression with the value 2. So, our expression (x + 4)² + x² + 16 becomes (2 + 4)² + 2² + 16. This is a crucial step because it sets the stage for the rest of the calculation. We've now transformed our algebraic expression into a purely numerical one.

  2. Parentheses: According to the order of operations, we need to tackle the parentheses first. Inside the parentheses, we have 2 + 4, which is a simple addition. 2 + 4 equals 6. So, our expression now looks like this: (6)² + 2² + 16.

  3. Exponents: Next up are the exponents. We have two terms with exponents: (6)² and 2². Remember, an exponent means we multiply the base number by itself the number of times indicated by the exponent. So, (6)² means 6 * 6, which equals 36. And 2² means 2 * 2, which equals 4. Now our expression is: 36 + 4 + 16.

  4. Addition: Finally, we're left with a series of additions. We simply add the numbers together from left to right. 36 + 4 equals 40. Then, 40 + 16 equals 56. So, the final result of our calculation is 56.

It's important to double-check each step to ensure accuracy. A small error in one step can throw off the entire solution. Now that we've carefully worked through the problem, we have our answer.

Identifying the Correct Answer

So, we've crunched the numbers and arrived at our solution: 56. But wait a minute! If we look back at the multiple-choice options provided in the question, we see: A) 36, B) 40, C) 44, and D) 48. None of these match our calculated answer of 56. What gives?

This is a crucial moment for reflection. It's tempting to panic or assume we've made a mistake, but the best approach is to calmly review our steps. Let's quickly recap what we did: We substituted x = 2 into the expression (x + 4)² + x² + 16, simplified the expression inside the parentheses, calculated the exponents, and then performed the addition. We followed the order of operations meticulously.

If we're confident in our steps, then the discrepancy between our answer and the provided options suggests there might be an error in the question itself or the answer choices. This happens sometimes in math problems, especially in practice questions or quizzes. It's a good reminder that even though math is precise, errors can occur.

In a real-world scenario, if you encountered this situation on a test, you might want to double-check your work and, if you're still confident in your answer, bring it to the attention of the instructor or test administrator. It's always better to be proactive and clarify any confusion.

For our purposes here, we've demonstrated the correct method for evaluating the expression. Our calculated answer is 56, and if the provided options don't include 56, then the correct answer is "None of the above" or the question might need to be reviewed.

Conclusion

Alright, guys, we've reached the end of our mathematical journey for this problem! We successfully evaluated the expression (x + 4)² + x² + 16 when x = 2. We walked through the steps carefully, emphasizing the importance of following the order of operations (PEMDAS/BODMAS) to ensure accuracy. We substituted the value of x, simplified the parentheses, calculated the exponents, and then added the terms together.

Our calculations led us to the answer 56. However, we also encountered a situation where our answer didn't match any of the multiple-choice options provided. This gave us an opportunity to discuss the importance of double-checking our work and recognizing that errors can sometimes occur in problem statements or answer keys.

The key takeaways from this exercise are:

  • Understanding the Order of Operations: PEMDAS/BODMAS is your best friend in math. It ensures you solve expressions in the correct sequence.
  • Step-by-Step Approach: Break down complex problems into smaller, manageable steps. This reduces the chance of errors.
  • Double-Checking Your Work: Always review your calculations to catch any mistakes.
  • Critical Thinking: If your answer doesn't match the given options, don't panic! Reassess the problem and your solution.

Math problems like these are excellent for honing our algebraic skills and critical thinking abilities. Remember, practice makes perfect, so keep tackling those expressions and equations! And most importantly, don't be afraid to ask questions and seek clarification when you need it. Math can be fun and rewarding when you approach it with a positive attitude and a willingness to learn.

So, until next time, keep those calculators handy and keep exploring the wonderful world of mathematics! You've got this!