Solve The Formula $A = X(100-2x)$: A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving into a fun little formula: $A = x(100 - 2x)$. Our mission? To fill out a table of values and see how this equation behaves. Don't worry, it's not as scary as it sounds. We'll break it down step by step, making sure everyone understands the process. This is a great way to brush up on your algebra skills and see how formulas work in the real world. So, grab your calculators (or your brains!) and let's get started. We're going to make this super easy to follow, so you'll be a pro in no time.

Understanding the Formula and Setting Up

Alright, first things first, let's understand the formula: $A = x(100 - 2x)$. This formula describes a relationship between a variable x and a value A. Think of x as the input, and A as the output. The formula tells us how to calculate A for any given value of x. The cool thing about formulas is that they help us model real-world situations. In this case, this particular formula can represent anything from the area of a rectangle to the profit of a business, depending on the context of x and A. In our table, we'll be plugging in different values for x and seeing what A turns out to be. It's like a mathematical treasure hunt – we provide the clues (x values), and the formula leads us to the treasure (A values). Make sure to understand that the order of operations is also important here. We first multiply x to the result of (100 - 2x). This is a common algebra concept, so understanding this will help you solve more complex equations down the road.

Now, let's look at the table. It has three rows: x, (100 - 2x), and A = x(100 - 2x). The x row will contain the input values, and the other two rows are where we'll do our calculations. Specifically, the second row calculates (100 - 2x), and the third row calculates A using our main formula. We’ll calculate them one at a time. This methodical approach is the key to getting accurate results. Using a table helps us organize the data and makes it easy to spot any patterns or trends. So, as you see, it's not just about getting the right answer; it's also about understanding how we get there. This systematic process is applicable to many other types of problems, so it's a valuable skill to learn. And that's why it is useful for everyone from students to professionals.

Before we begin, remember the basics of algebra. The most important thing is to be organized. Write down each step, even if it seems simple. And always double-check your work! This will help us avoid mistakes and make sure we have the correct values in our table. Remember, A is the result of the formula $A = x(100 - 2x)$. So, as you see, calculating A requires following the order of operations and plugging the input value into the formula correctly. In other words, our goal is to find the value of A by using the value of x and following the instructions in the formula. Make sure to follow the order of operations, and remember that any minor error can lead to a wrong answer. That's why it's very important to double-check your work and ensure accuracy.

Filling Out the Table: Step-by-Step Calculations

Let's start filling out the table. We'll go through each x value and calculate the corresponding values for (100 - 2x) and A. We will start with x = 0.

• When x = 0:
  • (100 - 2x) = 100 - 2(0) = 100 - 0 = 100
  • A = x(100 - 2x) = 0(100) = 0

Great! So, when x is 0, (100 - 2x) is 100, and A is 0. Next up, x = 5.

• When x = 5:
  • (100 - 2x) = 100 - 2(5) = 100 - 10 = 90
  • A = x(100 - 2x) = 5(90) = 450

See how it works? Let's keep going. For x = 10:

• When x = 10:
  • (100 - 2x) = 100 - 2(10) = 100 - 20 = 80
  • A = x(100 - 2x) = 10(80) = 800

Keep in mind that the calculation for A depends on the result of the calculation for (100 - 2x), so make sure to double-check your work and avoid mistakes. It’s also a good idea to perform these calculations on a piece of paper so that you can easily track your progress and reduce the chances of making mistakes. This method is the best way to ensure the accuracy of the final answers. And also, make sure to pay attention to details! Remember, that the goal is not only to find the correct answer but also to understand the calculations involved, so we can apply these steps to other problems. To reinforce our understanding, let’s solve the next one.

Next, let's calculate the values when x = 15:

• When x = 15:
  • (100 - 2x) = 100 - 2(15) = 100 - 30 = 70
  • A = x(100 - 2x) = 15(70) = 1050

So far, so good, right? Next is x = 20:

• When x = 20:
  • (100 - 2x) = 100 - 2(20) = 100 - 40 = 60
  • A = x(100 - 2x) = 20(60) = 1200

As you can see, the value of A increases as we increase the value of x. Also, the process remains the same! We are getting closer to completing the table and learning more about how this formula works. Let's keep the pace and calculate the values when x = 25:

• When x = 25:
  • (100 - 2x) = 100 - 2(25) = 100 - 50 = 50
  • A = x(100 - 2x) = 25(50) = 1250

Now, for x = 30:

• When x = 30:
  • (100 - 2x) = 100 - 2(30) = 100 - 60 = 40
  • A = x(100 - 2x) = 30(40) = 1200

And now, when x = 35:

• When x = 35:
  • (100 - 2x) = 100 - 2(35) = 100 - 70 = 30
  • A = x(100 - 2x) = 35(30) = 1050

Now let's compute for x = 40:

• When x = 40:
  • (100 - 2x) = 100 - 2(40) = 100 - 80 = 20
  • A = x(100 - 2x) = 40(20) = 800

We did it! We have completed all the calculations for the values provided. We have successfully calculated all the values. And it's as simple as that!

Completed Table of Values

Here is the completed table:

As you can see, the values of A increase, reach a maximum point, and then decrease. This kind of pattern is typical of quadratic equations. By working through this table, we've gained a better understanding of how the formula works and how A changes in relation to x. This can also be visualized as a parabola. This basic exercise is a stepping stone to understanding more complex mathematical concepts, so congratulations to you!

Final Thoughts and Next Steps

Awesome work, everyone! We successfully filled out the table and explored the behavior of the formula $A = x(100 - 2x)$. This is a fundamental concept in algebra, and understanding it can open doors to more advanced topics. Remember, practice is key. The more you work with formulas and equations, the more comfortable you'll become. So, keep practicing! You could try experimenting with different values of x or even explore other formulas. Perhaps you can use different values of x to see if you can get a better result. Math can be really exciting when you understand the logic behind it. This process can be applied to other formulas and is not limited to only one function.

If you enjoyed this, you might want to try other mathematical problems to strengthen your skills. Also, you can change the values of x and see how the result changes. Understanding the logic is more important than memorizing, so keep practicing and exploring! Thanks for joining me today. Keep up the great work! And remember, math can be fun!