Solve For G: Making The Equation True

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Solve for g: Making the Equation True

Hey guys! Let's dive into a cool math problem. We're going to figure out the value of g that makes the equation (x + 7)(x - 4) = x^2 + gx - 28 true. It's like a puzzle, and we're the detectives! We'll break it down step-by-step so it's super easy to understand. This is a common type of problem you might see in algebra, so understanding it will help you with a bunch of other math stuff. Ready to get started? Let's go!

Understanding the Problem

Okay, so what are we actually trying to do? We've got this equation, (x + 7)(x - 4) = x^2 + gx - 28. The left side of the equation is (x + 7)(x - 4), and the right side is x^2 + gx - 28. Our goal is to find the value of g that makes these two sides equal. Think of it like a balance scale; we want to make sure both sides weigh the same. The x is a variable, meaning it can be any number, but g is a specific number that we need to uncover. To do this, we'll need to expand the left side of the equation and then compare it to the right side. It's all about matching up the parts! Remember that the distributive property is your best friend here. This kind of problem helps us understand how variables and constants interact in algebraic expressions. This skill is critical when you get to more advanced topics. It is not scary, I promise. We will do this together.

Now, let's look at the equation again. See that (x + 7)(x - 4)? It means we need to multiply those two binomials together. Binomials are just fancy words for expressions with two terms. We'll use the distributive property (sometimes called the FOIL method, which stands for First, Outer, Inner, Last) to do this. Essentially, we multiply each term in the first binomial by each term in the second binomial. Here's how it looks:

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

So, after expanding, (x + 7)(x - 4) becomes x^2 - 4x + 7x - 28. See? Not too bad, right? We've taken something that looked a little complex and broken it down into smaller, easier pieces. Keep in mind that understanding the distributive property is really important. The distributive property is a fundamental concept in algebra, and it's used everywhere. Once you have it, you can move on to the next one. This step helps us simplify and manipulate the equation, making it easier to solve for g. Next, we'll combine the like terms.

Expanding and Simplifying the Equation

Alright, we've expanded (x + 7)(x - 4) to x^2 - 4x + 7x - 28. Now, we need to simplify it by combining like terms. What are like terms, you ask? They are terms that have the same variable raised to the same power. In our expanded expression, -4x and 7x are like terms because they both have x to the power of 1. To combine them, we just add their coefficients (the numbers in front of the x). So, -4 + 7 = 3. This means that -4x + 7x simplifies to 3x. Our equation now looks like this: x^2 + 3x - 28. Awesome! We're making progress. The goal is to get the equation in a form where we can directly compare it to x^2 + gx - 28. We're getting closer to solving for g. Remember, understanding how to combine like terms is a key skill in algebra. It helps us reduce the complexity of the equations, so we can isolate the variable we're interested in.

Let's recap what we did. We started with the expression (x + 7)(x - 4). We then used the distributive property, multiplying each term in the first parenthesis by each term in the second one. This gave us x^2 - 4x + 7x - 28. After that, we combined the like terms -4x and 7x, which simplified to 3x. So the simplified equation became x^2 + 3x - 28. This is the same as the left side of the original equation, expanded and simplified. We are now in a position to easily compare it to the right side of the original equation, x^2 + gx - 28. And that is what we are going to do next.

Solving for g

Now comes the fun part: solving for g! We've simplified the left side of the equation to x^2 + 3x - 28. The right side of the equation is x^2 + gx - 28. For the original equation to be true, the left side must equal the right side. That means the coefficients of the corresponding terms must be equal. Let's break this down further. Notice that both sides have x^2 and -28. Those parts match perfectly. The only difference is the middle term: on the left side, we have 3x, and on the right side, we have gx. This is where g comes in. Since the entire expressions must be equal, the coefficients of x must also be equal. That means that the coefficient of x on the left side, which is 3, must be equal to g. Boom! We've found our answer. Therefore, g = 3. See? Not so tough after all, right? The key here is understanding that for two expressions to be equal, their corresponding terms must have the same coefficients. This concept is fundamental to solving algebraic equations.

So, if we substitute g = 3 back into the original equation, we get (x + 7)(x - 4) = x^2 + 3x - 28. If you expand the left side, you'll see that it does indeed equal the right side. That’s how you know you have the correct answer. This is a good habit, always plug the solution back in to make sure that the original equation is true. This simple check can save you from a lot of silly mistakes. It is all about paying attention to the details. We successfully found the value of g by expanding, simplifying, and then comparing the coefficients of the x terms. Good job, guys!

Conclusion

So, to recap, the value of g that makes the equation (x + 7)(x - 4) = x^2 + gx - 28 true is g = 3. We arrived at this solution by expanding the left side of the equation, simplifying it, and then comparing the coefficients of the x terms on both sides. This problem illustrates the importance of understanding the distributive property, combining like terms, and recognizing how the coefficients of terms relate to each other in an equation. These are fundamental skills in algebra that will help you tackle more complex problems in the future. Keep practicing, and you'll become a math whiz in no time. If you have any questions or want to try some more problems, let me know! It's super fun.

I hope this explanation was helpful and easy to follow. Remember, math is like a puzzle, and it’s always rewarding to solve one. Keep up the good work, and keep exploring the amazing world of mathematics! You've got this!