Solving For X: (x+7)(x-4) = X^2 + 9x - 28

by Andrew McMorgan 42 views

Hey math enthusiasts! Ever stumbled upon an equation that looks a bit intimidating at first glance? Well, fear not! Today, we're diving into a fun and engaging problem where we'll find the value of 'x' that makes the equation (x+7)(x-4) = x^2 + 9x - 28 true. So, grab your thinking caps, and let's get started!

Unpacking the Equation

When faced with an equation like this, the first step is to understand what it's asking. In essence, we're looking for the number 'x' that, when plugged into the equation, will make both sides equal. It's like a puzzle where we need to find the missing piece that fits perfectly. Before we jump into solving, let's break down the equation piece by piece. On the left side, we have (x+7)(x-4), which means we're multiplying two expressions involving 'x'. This is a classic setup for using the distributive property, which we'll get to in a moment. On the right side, we have a quadratic expression: x^2 + 9x - 28. This tells us that we might end up with a quadratic equation to solve, but let's not get ahead of ourselves. Remember guys, the key to solving any math problem is to take it one step at a time. Don't let the complexity overwhelm you. Think of it as climbing a staircase; you focus on the next step, and eventually, you reach the top. The first step here is to simplify both sides of the equation as much as possible. This often involves expanding expressions and combining like terms. By simplifying, we make the equation easier to work with and reveal its underlying structure. It’s like decluttering a room; once everything is organized, you can see the space more clearly and move around more efficiently. So, let’s roll up our sleeves and start simplifying this equation. We'll begin by expanding the left side, which will give us a clearer picture of what we're dealing with. Trust me, once we've simplified things, the solution will be much easier to spot. And who knows, you might even enjoy the process! Remember, math isn't just about finding the right answer; it's about the journey of problem-solving and the satisfaction of cracking the code.

Step-by-Step Solution

Now, let's dive into the nitty-gritty of solving for 'x'. We'll break it down into manageable steps so you can follow along easily.

1. Expand the Left Side

The first order of business is to expand the left side of the equation, (x+7)(x-4). To do this, we'll use the distributive property, also known as the FOIL method (First, Outer, Inner, Last). This method ensures that each term in the first expression is multiplied by each term in the second expression. So, let's break it down:

  • First: x * x = x^2
  • Outer: x * -4 = -4x
  • Inner: 7 * x = 7x
  • Last: 7 * -4 = -28

Now, we add these terms together: x^2 - 4x + 7x - 28. Notice that we have two terms with 'x' in them (-4x and 7x), which we can combine. This gives us x^2 + 3x - 28. So, the expanded form of the left side of the equation is x^2 + 3x - 28. Remember guys, expanding expressions is a fundamental skill in algebra. It’s like knowing your scales in music; it’s essential for playing more complex pieces. And the more you practice, the more natural it becomes. Think of it as training your mathematical muscles; with each expansion, you become stronger and more confident in your abilities. And the beauty of math is that there's always a sense of accomplishment when you successfully expand an expression and see how it simplifies the equation. It's like solving a mini-puzzle within the larger problem. So, take your time, double-check your work, and enjoy the process of expanding and simplifying. It's a crucial step towards finding the value of 'x' and unlocking the solution to our equation.

2. Simplify the Equation

Now that we've expanded the left side, our equation looks like this: x^2 + 3x - 28 = x^2 + 9x - 28. The next step is to simplify the equation by getting all the terms on one side. This will help us isolate 'x' and find its value. To do this, we can start by subtracting x^2 from both sides of the equation. This eliminates the x^2 term from both sides, making the equation simpler. When we subtract x^2 from both sides, we get 3x - 28 = 9x - 28. Notice how much cleaner the equation looks now! This is the power of simplification. It's like weeding a garden; you remove the unnecessary clutter to allow the important plants to thrive. Now, let's move on to the next step. We want to get all the 'x' terms on one side and the constant terms on the other. To do this, we can subtract 3x from both sides of the equation. This will eliminate the 'x' term on the left side and move it to the right side. When we subtract 3x from both sides, we get -28 = 6x - 28. We're getting closer to isolating 'x'! The equation is becoming more and more manageable. It's like peeling an onion; with each layer you remove, you get closer to the core. Now, let's deal with the constant terms. We have -28 on both sides of the equation. To eliminate the -28 on the left side, we can add 28 to both sides. This will cancel out the -28 on the left and leave us with just the 'x' term on the right. When we add 28 to both sides, we get 0 = 6x. Wow! The equation has simplified beautifully. We're now just one step away from finding the value of 'x'. Remember, simplification is key in mathematics. It's like streamlining a process; you eliminate the unnecessary steps to make the task more efficient. And in this case, by simplifying the equation, we've made it much easier to solve for 'x'.

3. Isolate x

We've reached the final step! Our simplified equation is 0 = 6x. To isolate 'x', we need to get it by itself on one side of the equation. Since 'x' is being multiplied by 6, we can do the opposite operation: divide both sides of the equation by 6. When we divide both sides by 6, we get 0 / 6 = 6x / 6. This simplifies to 0 = x. And there you have it! We've found the value of 'x' that makes the equation true. It's like reaching the summit of a mountain after a challenging climb. The view from the top is always worth the effort. So, the value of 'x' that satisfies the equation (x+7)(x-4) = x^2 + 9x - 28 is x = 0. Isn't it satisfying when all the pieces of the puzzle come together? This final step of isolating 'x' is like the last stroke of a painter's brush; it completes the masterpiece. And in this case, our masterpiece is the solution to the equation. Remember guys, math is like a journey of discovery. Each step builds upon the previous one, leading you closer to the answer. And the satisfaction of finding the solution is a reward in itself. So, celebrate your success! You've successfully solved for 'x' and conquered the equation.

Verification

To ensure our answer is correct, let's substitute x = 0 back into the original equation and see if both sides are equal. This is like checking your work in any task; it's a crucial step to ensure accuracy and avoid mistakes. So, let's plug in x = 0 into the equation (x+7)(x-4) = x^2 + 9x - 28. On the left side, we have (0+7)(0-4), which simplifies to (7)(-4), which equals -28. On the right side, we have 0^2 + 9(0) - 28, which simplifies to 0 + 0 - 28, which also equals -28. Since both sides of the equation are equal when x = 0, we can confidently say that our solution is correct! This verification step is like a final confirmation; it gives you the peace of mind knowing that you've solved the problem accurately. It's like getting a thumbs up from a teacher after a test; it's a validation of your hard work and understanding. Remember, checking your work is a valuable habit in math and in life. It helps you catch errors and ensures that you're on the right track. So, always take the time to verify your solutions. It's a small step that can make a big difference in your accuracy and confidence.

Conclusion

And there you have it! We've successfully solved for 'x' in the equation (x+7)(x-4) = x^2 + 9x - 28 and found that x = 0. We walked through the process step-by-step, from expanding the left side to simplifying the equation and finally isolating 'x'. We even verified our answer to make sure it's correct. Solving equations like this is a fundamental skill in algebra, and it's something you'll use again and again in your math journey. It's like learning to ride a bike; once you get the hang of it, you can go anywhere. And the more you practice, the more confident and proficient you'll become. So, don't be afraid to tackle challenging equations. Break them down into smaller steps, use the tools you've learned, and enjoy the process of problem-solving. Remember, math isn't just about numbers and symbols; it's about critical thinking, logical reasoning, and the satisfaction of finding solutions. So, keep exploring, keep learning, and keep solving! And who knows, maybe you'll even discover a passion for math along the way. It's a world of endless possibilities, just waiting to be explored. Happy solving, guys!