Solving For 'a' In A Cubic Function: A Step-by-Step Guide

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Solving for 'a' in a Cubic Function: A Step-by-Step Guide

Hey guys! Let's dive into a cool math problem. We're given a cubic function, and we need to figure out the value of a specific variable within it. Sounds fun, right? Don't worry, it's not as scary as it might seem. We'll break it down step by step, making it super easy to follow along. So, grab your pencils (or your favorite note-taking app), and let's get started! This is a classic algebra problem, and understanding how to solve it is a fundamental skill. It not only helps with your math grades but also builds a strong foundation for more advanced concepts down the road. We'll explore the importance of substitution, the order of operations, and how to isolate a variable. By the end of this guide, you'll be able to confidently tackle similar problems. Let's make learning math enjoyable and accessible for everyone. Ready to unlock the mystery of this cubic function? Let's go!

Understanding the Problem: The Basics

Alright, so here's the deal: We've got a function, denoted as f(x). This function is a cubic function, which means the highest power of the variable x is 3. The function is defined as: f(x) = 2x³ - x² + ax - 5. Notice that there's an 'a' in there? That's the value we're trying to find! The problem also tells us that when x is equal to 2, the value of the function, f(2), is equal to 15. This is our key piece of information. Think of it this way: We have a machine (the function) that takes an input (x), does some calculations, and spits out an output (f(x)). We know what the input is in this specific case (2), and we know what the output is (15). Our mission is to use this information to determine the hidden value, a. Understanding the problem is half the battle. We need to clearly identify what we know, what we want to find, and how the different pieces of information relate to each other. This is crucial for formulating a plan to solve the problem. The ability to break down a complex problem into smaller, manageable parts is a valuable skill that applies not only in mathematics but also in various aspects of life. In this problem, we are looking for the 'a' value, which makes this problem unique and interesting.

Breaking it Down: What We Know and Need

Let's be absolutely clear about what we have:

  • The function: f(x) = 2x³ - x² + ax - 5
  • The specific value: f(2) = 15

Our goal is to find the value of 'a'. To do this, we'll use the given information f(2) = 15. This means that when we substitute x = 2 into the function, the result should be 15. This substitution is the cornerstone of our solution. By replacing x with 2, we transform the general function f(x) into a specific equation with a known result. This lets us focus on solving for the unknown, 'a'. This step is a critical application of the concept of function evaluation. This method is used a lot in mathematical analysis. Being comfortable with substitution and function evaluation is a must-have skill in algebra. Remember, in algebra, we often use letters (like a, x) to represent unknown values. Solving for these unknowns is the essence of algebra, and it's a skill you'll use throughout your math journey. So, let's substitute and conquer!

The Substitution Step: Plugging in the Value

Okay, time for the fun part: substitution! We're going to replace every x in the function with the value 2. Let's do it step by step to avoid any mistakes.

  • Original function: f(x) = 2x³ - x² + ax - 5
  • Substitute x = 2: f(2) = 2(2)³ - (2)² + a(2) - 5

See? We've replaced every x with a 2. Now we can simplify this expression. This is where we apply the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Always follow PEMDAS or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) to ensure we evaluate the equation correctly. This is very important. Without a proper application of the order of operations, we'll end up with the wrong answer. Keep the operations correct to successfully solve this kind of problem. In this step, we've replaced variables with actual numbers.

Simplifying the Expression

Let's simplify that expression we just created: f(2) = 2(2)³ - (2)² + a(2) - 5

  1. Exponents: First, we calculate the exponents:

    • 2³ = 2 * 2 * 2 = 8
    • 2² = 2 * 2 = 4 So our equation becomes: f(2) = 2(8) - 4 + 2a - 5

  2. Multiplication: Next, we handle the multiplication:

    • 2(8) = 16 Our equation is now: f(2) = 16 - 4 + 2a - 5

  3. Combine Like Terms: Finally, let's combine the constant terms (the numbers without any variables):

    • 16 - 4 - 5 = 7

So our simplified equation is: f(2) = 7 + 2a

Remember, we also know that f(2) = 15. Therefore, we can set up the following equation: 15 = 7 + 2a. The equation we have now is in a form that we can solve! We have one variable and several numbers, we can now solve for our target 'a'.

Isolating 'a': Solving for the Unknown

We're almost there! We've simplified the equation to 15 = 7 + 2a. Now, our goal is to isolate 'a' on one side of the equation. This means getting 'a' by itself. Here's how we do it, step-by-step.

  1. Subtract 7 from both sides: To get rid of the 7 on the right side, we subtract 7 from both sides of the equation. Remember, whatever we do to one side of the equation, we must do to the other side to keep it balanced.
    • 15 - 7 = 7 + 2a - 7
    • This simplifies to: 8 = 2a
  2. Divide both sides by 2: Now we need to get rid of the 2 that's multiplying 'a'. To do this, we divide both sides of the equation by 2.
    • 8 / 2 = 2a / 2
    • This simplifies to: 4 = a

Therefore, we have found that a = 4!

Checking the Answer: Verification is Key

It's always a good idea to check your answer! Let's substitute a = 4 back into the original function and see if f(2) equals 15.

  • Original function with a = 4: f(x) = 2x³ - x² + 4x - 5
  • Substitute x = 2: f(2) = 2(2)³ - (2)² + 4(2) - 5
  • Simplify: f(2) = 2(8) - 4 + 8 - 5
  • Calculate: f(2) = 16 - 4 + 8 - 5 = 15

Success! When a = 4 and x = 2, f(2) indeed equals 15. This confirms that our solution is correct. This step is super important, as it helps identify any arithmetic errors we might have made along the way. Verification is an essential part of problem-solving. This kind of problem is just a puzzle, and verifying our solution is like confirming that we have the right puzzle pieces.

Conclusion: You Did It!

Awesome work, guys! We successfully found the value of 'a' in our cubic function. We went from a problem with an unknown to a clear solution. We have successfully solved the problem. Remember, we used substitution, followed the order of operations, and isolated the variable. These skills are fundamental to algebra and will serve you well in future math problems. The key takeaways from this problem are:

  • Understand the problem: Always clearly identify what you know and what you need to find.
  • Substitution: Replace variables with their given values.
  • Order of Operations (PEMDAS/BODMAS): Follow the correct order to simplify expressions.
  • Isolate the variable: Use algebraic operations to get the variable by itself.
  • Verify your answer: Always double-check your work to ensure accuracy.

Keep practicing, and you'll become a pro at these types of problems in no time. Congratulations on conquering this math challenge! Keep up the great work, and don't be afraid to keep practicing. Math can be fun when we approach it with the right mindset and a willingness to learn. Keep exploring and challenging yourselves, and you'll discover the amazing power and beauty of mathematics!