Missing Side Length: Solving (x-13)(x+8)=196

Hey guys! Let's dive into a fun math problem today where we're figuring out the missing side length of a rectangle. The equation we're working with is (x-13)(x+8)=196, and the side length we're trying to find is represented by x+8. It might look a bit intimidating at first, but trust me, we'll break it down step by step so it's super clear. So, grab your thinking caps, and let's get started!

Understanding the Problem

Okay, so the first thing we need to do is really understand what the problem is asking. We've got this rectangle, right? And we know that one of its sides can be described as x+8 units long. The equation (x-13)(x+8)=196 is like a secret code that holds the key to finding out what x is. Once we know x, we can easily figure out the length of that missing side. Think of it like a puzzle – each piece of information fits together to give us the final answer. It's important to visualize the rectangle and understand that the equation relates to its area, which is 196 square units. This equation is derived from the formula for the area of a rectangle, which is length × width. In our case, (x-13) represents one side, (x+8) represents the other side, and their product gives us the area.

To really nail this, let's think about what each part of the equation means. The left side, (x-13)(x+8), is telling us how to calculate the area of the rectangle if we knew the value of x. The right side, 196, is the actual area of the rectangle. Our mission is to find the value of x that makes the left side equal to the right side. This is a classic algebra problem, and we're going to use our skills to solve it. So, with that in mind, let’s move on to the next step where we’ll actually start solving the equation. Remember, math is like a story – each step leads us closer to the exciting conclusion!

Solving the Quadratic Equation

Alright, let's get our hands dirty with some algebra! The equation we need to solve is (x-13)(x+8)=196. The first thing we want to do is expand the left side of the equation. We're going to use the FOIL method, which stands for First, Outer, Inner, Last. This helps us make sure we multiply each term in the first set of parentheses by each term in the second set. So, let's break it down:

• First: x times x equals x²
• Outer: x times 8 equals 8x
• Inner: -13 times x equals -13x
• Last: -13 times 8 equals -104

Now, we combine these terms: x² + 8x - 13x - 104. We can simplify this further by combining the like terms (the ones with x): x² - 5x - 104. So, our equation now looks like this: x² - 5x - 104 = 196. We're getting somewhere!

But we're not done yet. To solve this quadratic equation, we need to set it equal to zero. We can do this by subtracting 196 from both sides of the equation: x² - 5x - 104 - 196 = 0. This simplifies to x² - 5x - 300 = 0. Now we have a classic quadratic equation in the form of ax² + bx + c = 0. The next step is to factor this equation. Factoring means finding two numbers that multiply to -300 and add up to -5. This might take a little bit of trial and error, but don't worry, we'll get there! Think about pairs of factors of 300 and see which ones have a difference of 5. After a bit of thought, we'll find that the numbers are -20 and 15 because -20 * 15 = -300 and -20 + 15 = -5. So, we can rewrite the equation as (x - 20)(x + 15) = 0.

Now we're in the home stretch! To find the values of x, we set each factor equal to zero: x - 20 = 0 and x + 15 = 0. Solving these simple equations gives us two possible values for x: x = 20 and x = -15. But hold on, we're talking about the side length of a rectangle, and lengths can't be negative. So, we can discard the solution x = -15. That leaves us with x = 20. Woo-hoo! We've found x! Now, let’s use this value to find the missing side length.

Finding the Missing Side Length

Okay, we've done the heavy lifting and found that x = 20. Remember, the side length we're trying to find is represented by x + 8. So, all we need to do is substitute the value of x into this expression. This means we replace x with 20: 20 + 8. And that's a simple addition problem! 20 + 8 = 28.

So, the missing side length of the rectangle is 28 units. Awesome! We've solved the puzzle. Let's quickly recap what we did to make sure we've got it all clear. First, we expanded the original equation using the FOIL method. Then, we simplified it and set it equal to zero to get a quadratic equation. We factored the quadratic equation and found two possible values for x. We discarded the negative value because a side length can't be negative, and we were left with x = 20. Finally, we substituted this value into the expression x + 8 to find the missing side length, which is 28 units. See? It wasn't so scary after all!

Now, let's go back to those answer choices and see which one matches our solution. We've got A. 7, B. 15, C. 20, and D. 28. Aha! Our answer, 28, matches option D. So, we can confidently say that the missing side length of the rectangle is 28 units. High five! You've just tackled a pretty cool algebra problem. Remember, math is all about breaking things down into smaller, manageable steps. Keep practicing, and you'll become a math whiz in no time!

Conclusion and Key Takeaways

So, guys, we've successfully navigated through this problem and found the missing side length of the rectangle. We started with the equation (x-13)(x+8)=196 and ended up with the answer 28 units. That's quite a journey! Let's take a moment to reflect on the key takeaways from this problem. First and foremost, we saw how important it is to understand the problem before diving into the calculations. We needed to recognize that the equation represented the area of a rectangle and that x+8 was one of its sides. This understanding laid the foundation for our entire solution.

Next, we practiced our algebra skills. We used the FOIL method to expand the equation, simplified it, and then rearranged it into a quadratic equation. Remember, the FOIL method is a super handy tool for multiplying binomials (expressions with two terms), and it's something you'll use a lot in algebra. Then, we factored the quadratic equation. Factoring can be a bit tricky at first, but with practice, you'll get the hang of it. Look for those magic numbers that multiply to one thing and add up to another – it's like solving a mini-puzzle within the bigger puzzle.

We also learned about the importance of checking our solutions. We found two possible values for x, but one of them was negative. Since lengths can't be negative, we knew we had to discard that solution. Always remember to think about the context of the problem and make sure your answers make sense. It's a great way to catch mistakes and ensure your solution is correct. Finally, we saw how substitution works. Once we found the value of x, we simply plugged it into the expression x+8 to find the missing side length. Substitution is a fundamental concept in algebra, and you'll use it in many different types of problems.

In conclusion, problems like these are not just about finding the right answer; they're about building your problem-solving skills. Each step we took – understanding the problem, using the FOIL method, factoring, checking solutions, and substituting – is a valuable skill that you can apply to other math problems and even to real-world situations. So, keep practicing, keep asking questions, and keep challenging yourselves. You guys are awesome, and I know you can tackle any math problem that comes your way! Keep shining and see you in the next math adventure!