Evaluate 2(x-4) + 3x - X² For X = 4

by Kenji Nakamura 36 views

Hey everyone! Today, we're diving into a fun little math problem. We need to evaluate the expression 2(x-4) + 3x - x² when x = 4. Sounds like a mouthful, right? But don't worry, we'll break it down step-by-step so it's super easy to follow. We'll explore the fundamental concepts, then walk through the calculation, and finally land on the correct answer. So, let's get started and unravel this math puzzle together!

Understanding the Expression

Before we jump into plugging in the value of x, let's take a moment to understand what the expression 2(x-4) + 3x - x² actually means. This is a polynomial expression, which basically means it's a combination of variables (like 'x') and constants (like 2, 3, and 4) connected by mathematical operations like addition, subtraction, and multiplication. The exponent (the little '2' in x²) indicates that we're squaring the variable x.

The order of operations is super important here. Remember PEMDAS/BODMAS? It stands for:

  • Parentheses/Brackets
  • Exponents/Orders
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)

This tells us the order in which we need to perform the operations to get the correct result. First, we'll deal with any operations inside parentheses, then exponents, then multiplication and division, and finally addition and subtraction. It's like following a recipe – you need to add the ingredients in the right order to bake a delicious cake!

In our expression, we have parentheses (x-4), multiplication (2(x-4) and 3x), addition, subtraction, and an exponent (x²). Keeping PEMDAS/BODMAS in mind will help us navigate the calculation smoothly.

Step-by-Step Calculation

Alright, now for the fun part – plugging in x = 4 and crunching the numbers! Here’s how we’ll do it step by step:

  1. Substitute x with 4:

    Our expression is 2(x-4) + 3x - x². When we substitute x with 4, it becomes 2(4-4) + 3(4) - (4²).

    See? We've just replaced every 'x' with the number '4'. Easy peasy!

  2. Solve the parentheses:

    According to PEMDAS/BODMAS, we tackle the parentheses first. Inside the parentheses, we have (4-4), which equals 0. So, our expression now looks like this: 2(0) + 3(4) - (4²).

  3. Evaluate the exponent:

    Next up is the exponent. We have (4²), which means 4 * 4, which equals 16. Our expression is now: 2(0) + 3(4) - 16.

  4. Perform the multiplications:

    Now we handle the multiplications from left to right. First, we have 2(0), which equals 0. Then, we have 3(4), which equals 12. Our expression is now: 0 + 12 - 16.

  5. Do the addition and subtraction:

    Finally, we perform addition and subtraction from left to right. First, 0 + 12 equals 12. Then, 12 - 16 equals -4. So, our final result is -4!

    Wohoo! We did it! By following the order of operations and carefully substituting the value of x, we've successfully evaluated the expression.

Arriving at the Correct Answer

So, after our step-by-step calculation, we found that when x = 4, the expression 2(x-4) + 3x - x² equals -4. Let's take a look at the answer choices provided:

  • A. 8
  • B. -4
  • C. -6
  • D. -12

As you can see, the correct answer is B. -4. We nailed it!

It's always a good idea to double-check your work, especially in math problems. You can quickly run through the steps again to ensure you haven't made any silly mistakes. Accuracy is key!

Why is this important?

You might be thinking, “Okay, I can evaluate this expression, but why does it even matter?” Well, evaluating expressions like this is a fundamental skill in algebra and mathematics in general. It's like learning the ABCs of math – you need it to build more complex concepts.

Here are a few reasons why evaluating expressions is important:

  • Solving Equations: Evaluating expressions is a key step in solving equations. When you need to find the value of a variable that makes an equation true, you often need to evaluate expressions on both sides of the equation.
  • Graphing Functions: When you graph a function, you're essentially plotting points that result from evaluating the function for different values of the variable. Understanding how to evaluate expressions is crucial for understanding graphs.
  • Modeling Real-World Situations: Math is used to model real-world situations all the time, from calculating the trajectory of a rocket to predicting the growth of a population. These models often involve expressions that need to be evaluated.
  • Computer Programming: If you're interested in computer programming, you'll be evaluating expressions all the time. Programming languages use expressions to perform calculations and make decisions.

So, by mastering the skill of evaluating expressions, you're setting yourself up for success in more advanced math topics and even in other fields! It's a foundational skill that opens doors to many possibilities.

Tips and Tricks for Evaluating Expressions

Evaluating expressions can become second nature with practice. Here are a few tips and tricks to help you become a pro:

  • Always follow the order of operations (PEMDAS/BODMAS): This is the golden rule of evaluating expressions. Stick to it, and you'll avoid many common mistakes.
  • Write down each step: Don't try to do everything in your head. Writing down each step helps you keep track of your work and makes it easier to spot errors.
  • Double-check your work: It's always a good idea to go back and check your calculations, especially if you're working on a test or an important problem.
  • Practice, practice, practice: The more you practice evaluating expressions, the better you'll become at it. Try working through different types of expressions with varying levels of complexity.
  • Use online calculators or tools to check your answers: There are many online calculators and tools that can help you check your work. This can be a great way to build confidence and identify areas where you might be making mistakes.

Conclusion

Great job, everyone! We've successfully evaluated the expression 2(x-4) + 3x - x² for x = 4 and found the answer to be -4. We walked through the steps, discussed the importance of the order of operations, and even explored why evaluating expressions is a valuable skill in mathematics and beyond.

Remember, math is like a puzzle – each piece fits together to create a beautiful solution. Keep practicing, keep exploring, and most importantly, keep having fun with math! You guys are awesome!

If you have any questions or want to try more examples, feel free to ask. Keep up the great work!