Solving @Lio 9 X² A Mathematical Exploration Of Equations And Volume

Hey guys! Ever stumbled upon a math problem that looks like a crazy code? Well, that’s how the equation “@Lio:9 - x² menghasilkan y= x+79 - x² = x + 7” felt at first glance! But don't worry, we're going to break it down step by step, solve it, and even explore the awesome mathematical concepts hiding inside. Buckle up, because we're about to dive into the world of algebra and calculus, making it fun and easy to understand.

Unraveling the Equation: x + 79 - x² = x + 7

Let's start with the heart of the problem: the equation x + 79 - x² = x + 7. At first, it might seem like a jumble of numbers and variables, but trust me, there's a logical way to tackle it. Our main goal here is to find the value(s) of 'x' that make this equation true. To do that, we'll use some basic algebraic techniques. Think of it like solving a puzzle – we need to rearrange the pieces until we see the full picture.

First things first, let's simplify the equation. We can start by subtracting 'x' from both sides. Why? Because it cancels out the 'x' on both sides of the equals sign, making the equation simpler to deal with. This gives us: 79 - x² = 7. See? Much cleaner already!

Now, we want to isolate the x² term. To do this, let's subtract 79 from both sides of the equation. This leaves us with -x² = 7 - 79, which simplifies to -x² = -72. We're getting closer!

But we're not quite there yet. We need to get rid of that negative sign in front of the x². We can do this by multiplying both sides of the equation by -1. This gives us x² = 72. Now we're talking!

So, what does x² = 72 mean? It means we're looking for a number that, when multiplied by itself, equals 72. To find 'x', we need to take the square root of both sides of the equation. Remember, though, that there are two possible solutions: a positive square root and a negative square root. The square root of 72 is approximately 8.49. So, x ≈ 8.49 or x ≈ -8.49. We've found our x values!

Finding the Roots: x = 1 and x = -2

Okay, so we've tackled the first part of the equation. But wait, there's more! The original problem also mentions “Maka x = 1 dan x = -2”. This means we need to figure out how these values fit into the bigger picture. It looks like there might be another equation or condition at play here. Perhaps these are solutions to a related equation, or maybe they're critical points in a function. To understand this better, let's consider that these values might be the result of solving a quadratic equation, which often has two solutions or roots.

If x = 1 and x = -2 are roots, then we can work backward to find the quadratic equation they came from. Remember that if 'r' is a root of an equation, then (x - r) is a factor of that equation. So, if x = 1 is a root, then (x - 1) is a factor. And if x = -2 is a root, then (x + 2) is a factor. To find the quadratic equation, we can multiply these factors together: (x - 1)(x + 2) = 0. When we expand this, we get x² + 2x - x - 2 = 0, which simplifies to x² + x - 2 = 0. Ah-ha! Now we have a quadratic equation that has x = 1 and x = -2 as solutions. This is an important piece of the puzzle!

Delving into Volume Calculation: V = π∫[1,-2]((9-x²)² - (x + 7)²)dx

Now, let's talk about the next part of the problem: V = π∫[1,-2]((9-x²)² - (x + 7)²)dx. This looks like a volume calculation using integration, which is a concept from calculus. Don't let the symbols scare you – we'll break it down. The 'V' stands for volume, 'π' (pi) is the famous mathematical constant (approximately 3.14159), '∫' is the integral symbol (which basically means we're finding the area under a curve), and 'dx' tells us that we're integrating with respect to 'x'. The numbers 1 and -2 are the limits of integration, meaning we're finding the volume between x = -2 and x = 1. This formula suggests we are calculating the volume of a solid of revolution, where the solid is formed by rotating the area between two curves around an axis.

The expression ((9-x²)² - (x + 7)²) represents the difference between the squares of two functions: (9 - x²) and (x + 7). These functions define the curves that bound the area we're rotating. Think of it like this: we have two curves, and the volume we're calculating is the 3D shape formed when the area between those curves is spun around an axis (likely the x-axis in this case).

To calculate this volume, we first need to expand the expression inside the integral. Let's start with (9 - x²)². This means (9 - x²)(9 - x²), which expands to 81 - 18x² + x⁴. Next, let's expand (x + 7)². This means (x + 7)(x + 7), which expands to x² + 14x + 49. Now we can substitute these expansions back into the integral: V = π∫[1,-2](81 - 18x² + x⁴ - (x² + 14x + 49))dx. We're getting closer to solving this!

Simplifying and Integrating: Step-by-Step

Before we can integrate, we need to simplify the expression inside the integral. Let's distribute the negative sign and combine like terms: V = π∫[1,-2](81 - 18x² + x⁴ - x² - 14x - 49)dx. Combining like terms gives us: V = π∫[1,-2](x⁴ - 19x² - 14x + 32)dx. This looks much more manageable!

Now comes the fun part: integration! Remember, the integral of xⁿ is (x^(n+1))/(n+1). So, we'll apply this rule to each term in the expression. The integral of x⁴ is (x⁵)/5. The integral of -19x² is (-19x³)/3. The integral of -14x is (-14x²)/2, which simplifies to -7x². And the integral of 32 is 32x. So, the integral of the entire expression is (x⁵)/5 - (19x³)/3 - 7x² + 32x.

Now we need to evaluate this integral at the limits of integration, 1 and -2. This means we'll plug in x = 1 and x = -2 into the integrated expression and subtract the results. Let's call our integrated expression F(x) = (x⁵)/5 - (19x³)/3 - 7x² + 32x. We need to find F(1) and F(-2).

F(1) = (1⁵)/5 - (19(1)³)/3 - 7(1)² + 32(1) = 1/5 - 19/3 - 7 + 32. To add these fractions and whole numbers, we need a common denominator, which is 15. So, F(1) = 3/15 - 95/15 - 105/15 + 480/15 = 283/15.

Now let's find F(-2): F(-2) = ((-2)⁵)/5 - (19(-2)³)/3 - 7(-2)² + 32(-2) = -32/5 - (19(-8))/3 - 7(4) - 64 = -32/5 + 152/3 - 28 - 64. Again, we need a common denominator, which is 15. So, F(-2) = -96/15 + 760/15 - 420/15 - 960/15 = -616/15.

Finally, we subtract F(-2) from F(1): V = π(F(1) - F(-2)) = π(283/15 - (-616/15)) = π(283/15 + 616/15) = π(899/15). So, the volume V is approximately (899/15)π cubic units. We did it!

The Final Calculation: Putting It All Together

We've gone through a pretty amazing journey, guys! We started with a seemingly complex equation, broke it down step by step, and used algebra and calculus to solve it. We found the roots of a quadratic equation, understood the concept of volume calculation using integration, and even performed the integration and evaluation ourselves. That's some serious math power!

To recap, we found the values of 'x' that satisfy the equation, identified the quadratic equation associated with roots x = 1 and x = -2, and calculated the volume of the solid of revolution. The final answer for the volume is approximately (899/15)π cubic units. Remember, math isn't just about numbers and formulas – it's about problem-solving, logical thinking, and the thrill of discovery. Keep exploring, keep questioning, and keep having fun with math!

By understanding these steps, we've not only solved the problem but also gained a deeper appreciation for the beauty and power of mathematics. Keep practicing, and you'll be amazed at what you can achieve!