Finding The Value Of 'g' In The Equation (x+7)(x-4)

by Andrew McMorgan 52 views

Hey guys! Today, we're diving into a fun math problem where we need to figure out the value of 'g' that makes a certain equation true. Specifically, we're looking at the equation (x+7)(x-4) = x^2 + gx - 28. Sounds a bit like a puzzle, right? Let's break it down step by step so we can solve it together. This is a classic algebra problem that tests your understanding of expanding expressions and comparing coefficients. So, grab your thinking caps, and let’s get started!

Understanding the Problem

At the heart of this problem is the need to find the correct value for 'g' that will make both sides of the equation equal for any value of 'x'. This means we need to manipulate the equation, expand any expressions, and then compare the coefficients of the terms on both sides. Think of it like balancing a scale; we need to ensure both sides weigh the same. In the equation (x+7)(x-4) = x^2 + gx - 28, the left side is a product of two binomials, and the right side is a quadratic expression. Our mission, should we choose to accept it (and we do!), is to find the 'g' that makes this mathematical relationship harmonious. We will achieve this by first expanding the left side and then comparing it with the right side. This process will reveal the numerical value of 'g', giving us the solution to our algebraic quest. Remember, in algebra, each term and coefficient plays a critical role, and understanding these roles is key to solving equations effectively. So let’s roll up our sleeves and get to work!

Expanding the Left Side

Our first mission is to expand the left side of the equation, which is (x+7)(x-4). To do this, we'll use the FOIL method (First, Outer, Inner, Last), which is a handy way to multiply two binomials. Let's break it down:

  • First: Multiply the first terms in each binomial: x * x* = x^2
  • Outer: Multiply the outer terms: x * (-4)* = -4x
  • Inner: Multiply the inner terms: 7 * x = 7x
  • Last: Multiply the last terms: 7 * (-4) = -28

Now, let's put it all together: x^2 - 4x + 7x - 28. Next, we combine the like terms, which are -4x and 7x. Adding these together gives us 3x. So, the expanded form of the left side is x^2 + 3x - 28. This step is crucial because it transforms the product of binomials into a standard quadratic expression, making it easier to compare with the right side of the original equation. Mastering the FOIL method is super important in algebra, as it helps us simplify complex expressions and solve equations more efficiently. So, with the left side expanded, we’re one step closer to cracking this problem wide open. Onward to the next step!

Comparing Coefficients

Now that we've expanded the left side of the equation, we have x^2 + 3x - 28. Our original equation is (x+7)(x-4) = x^2 + gx - 28, which now looks like x^2 + 3x - 28 = x^2 + gx - 28. To find the value of 'g', we need to compare the coefficients of the x terms on both sides of the equation. Remember, coefficients are the numbers that multiply the variables. On the left side, the coefficient of x is 3, and on the right side, the coefficient of x is 'g'. For the equation to be true, these coefficients must be equal. So, we can directly equate them: g = 3. This comparison is a fundamental technique in algebra, allowing us to solve for unknowns by matching corresponding terms in equivalent expressions. It's like finding the missing piece of a puzzle by seeing where it fits perfectly. By comparing coefficients, we've zoomed in on the value of 'g' and found that it's 3. This method is not just useful for solving equations but also for understanding the structure and relationships within algebraic expressions. With 'g' identified, we’re nearing the finish line. Let's confirm our answer and wrap things up!

Solution

So, after expanding the left side of the equation and comparing the coefficients, we've found that the value of g that makes the equation (x+7)(x-4) = x^2 + gx - 28 true is g = 3. This means that if we substitute 3 for 'g' in the original equation, both sides will be equal for any value of x. To double-check, we can plug g = 3 back into the equation: x^2 + 3x - 28. This matches the expanded form of the left side, which we calculated earlier. Therefore, we can confidently say that our solution is correct. This entire process showcases the power of algebraic manipulation and the importance of understanding how equations work. By expanding, simplifying, and comparing, we turned a potentially confusing problem into a straightforward solution. Great job, guys! We tackled this math puzzle like pros. Now, let's celebrate this victory and keep our minds sharp for the next challenge. You've got this!

Choosing the Correct Option

Looking back at the options provided, we have:

  • A. -11
  • B. -3
  • C. 3
  • D. 11

Since we've determined that g = 3, the correct answer is C. 3. It's always a good feeling when your calculated answer matches one of the options, right? This confirms that our step-by-step approach was accurate and that we've correctly applied the principles of algebra. Selecting the right option from a set of choices is a crucial skill in mathematics, as it often appears in standardized tests and exams. By carefully working through the problem and verifying our solution, we can be confident in our choice and move forward with assurance. This exercise not only enhances our problem-solving abilities but also reinforces our understanding of algebraic concepts. So, kudos to us for nailing this one! We're one step closer to mastering the math universe, and I'm super excited about our progress. Keep up the awesome work!