Math Problem: Finding The Value Of 'b' In A Function

Hey Plastik Magazine readers! Ever stumbled upon a math problem that seems like a puzzle? Well, buckle up, because today we're diving deep into a function problem. We're going to break down how to find the value of 'b' when dealing with functions like f(x), g(x), and h(x). It's like a secret code, but instead of secret agents, we have equations! Let's get started, shall we?

Understanding the Core Functions

Alright, guys, let's look at the basic functions first. We have three main players in this game: f(x), g(x), and h(x). They're defined as follows:

• f(x) = 3x - 6
• g(x) = 2x + 4
• h(x) = 7x + 9

These are simple linear functions, meaning they'll create straight lines when you graph them. The cool part is, we're not just looking at them individually; we're going to mix them up like a mathematical cocktail! Our mission is to figure out the value of 'b' in the equation g(x) - f(x) * h(x) = ax² + bx + c. This equation involves these functions and asks us to find the value of 'b' after a little mathematical manipulation. So, let’s begin!

This is a classic algebra problem that involves substituting the functions and simplifying the resulting expression. The key here is to carefully perform the operations, paying close attention to the order of operations and the signs. If we mess up any of these, we will not get the correct value of 'b'. So, we will approach this carefully, step by step, which will help us solve the problem correctly. Ready to see the magic happen?

Step 1: Combining the Functions

The most important part of solving the question is substituting the functions into the equation. The equation we are working with is g(x) - f(x) * h(x) = ax² + bx + c. Our first move is to substitute the given functions into the left side of the equation. So, we'll replace g(x), f(x), and h(x) with their respective expressions. It becomes:

(2x + 4) - (3x - 6) * (7x + 9)

See? It's like replacing characters in a play. Now, the fun begins with the algebra. We'll start by multiplying f(x) and h(x).

Step 2: Multiplying f(x) and h(x)

Now, let's get our hands dirty by multiplying (3x - 6) and (7x + 9). This is where we need to remember the distributive property. Each term in the first set of parentheses must be multiplied by each term in the second set. It goes like this:

• 3x * 7x = 21x²
• 3x * 9 = 27x
• -6 * 7x = -42x
• -6 * 9 = -54

When we put it all together, (3x - 6) * (7x + 9) = 21x² + 27x - 42x - 54. Simplify this expression by combining like terms, which gives us 21x² - 15x - 54. So we are getting closer to our final answer. Good job guys!

Step 3: Completing the Calculation

Now that we've found the product of f(x) and h(x), we'll slot it back into our main equation. Remember, our original equation was (2x + 4) - (3x - 6) * (7x + 9) = ax² + bx + c. We now know that (3x - 6) * (7x + 9) = 21x² - 15x - 54. So, substitute this back into the equation. The equation then becomes:

(2x + 4) - (21x² - 15x - 54)

Now, we need to subtract the second set of parentheses from the first. Remember to distribute the negative sign! It's super important. This means every term inside the second set of parentheses changes its sign.

• 2x + 4 - 21x² + 15x + 54

Step 4: Simplifying and Finding 'b'

Great work, fellas! Now, all we have to do is simplify by combining like terms. Let’s organize our equation. We rearrange it to get:

-21x² + (2x + 15x) + (4 + 54)

Combine the terms:

-21x² + 17x + 58.

Now, we compare this with ax² + bx + c. It's pretty clear now, isn't it? We can directly match the coefficients. So, a = -21, b = 17, and c = 58. Ta-da! We've cracked the code! The value of b is 17. The question is solved, guys.

The Final Answer and Why It Matters

So, there you have it, folks! Through a little bit of algebraic manipulation, we successfully found that the value of b in the equation is 17. This is a great example of how mathematical concepts are interconnected. It's not just about memorizing formulas; it's about understanding how they fit together. This type of problem builds a foundation for more complex mathematical concepts and sharpens problem-solving skills.

Recap: Key Steps to Success

Just to recap, here's the game plan we followed:

1. Defined: We were given the functions f(x), g(x), and h(x).
2. Substituted: Plugged the function expressions into the equation.
3. Multiplied: Multiplied f(x) and h(x).
4. Simplified: Combined like terms to get a simplified quadratic equation.
5. Identified: Matched the coefficients to find the value of b.

This method can be applied to many similar problems. The key is to be organized, methodical, and careful with your calculations. Awesome, right?

Conclusion: Keep Practicing!

Well, that was fun, wasn't it? Hopefully, this helps you understand the process of solving such problems. Remember, practice makes perfect! The more you work through these types of problems, the easier they become. Keep exploring, keep questioning, and keep having fun with math! Until next time, stay curious!