Solving 6x² - 1 = Value Finding 36x² + 1/x²
Hey everyone! Today, we're diving into an exciting math problem where we'll solve an equation and then use that solution to find the value of another expression. Specifically, we're tackling the equation 6x² - 1 = hola and aiming to determine the value of 36x² + 1/x². This type of problem is fantastic because it combines algebraic manipulation with a bit of clever thinking. So, let's jump right in and break it down step by step!
Understanding the Problem
Before we start crunching numbers, let's make sure we fully grasp what the problem is asking. We're given an equation, 6x² - 1 = hola, and our ultimate goal is to find the value of the expression 36x² + 1/x². The key here is to recognize that these two expressions are related, and we can use the information from the first equation to help us with the second. It's like a puzzle where the first part gives us clues to solve the second part. Now, the "hola" might seem a bit out of place, right? Usually, we'd expect a number there. Let's assume for a moment that "hola" is just a placeholder for a number, and we'll need to figure out what that number should be to make the problem solvable. This kind of problem-solving often involves making educated guesses and seeing where they lead us. Think of it as mathematical detective work!
Initial Steps and Considerations
First, let's focus on the relationship between 6x² in the given equation and 36x² in the expression we want to find. Notice that 36x² is simply the square of 6x. This is a crucial observation because it suggests that squaring the entire equation 6x² - 1 = hola might be a useful strategy. Squaring equations can often reveal hidden connections and lead us closer to our goal. However, we also need to keep in mind the 1/x² term in the expression we're trying to evaluate. This term involves the reciprocal of x², which means we might need to manipulate the original equation to isolate x² or find a way to introduce its reciprocal. So, our initial plan is shaping up: we'll likely square the equation, but we also need to think about how to incorporate the reciprocal term. This is where the problem starts to get interesting, and we'll need to use our algebraic skills to navigate through it. Remember, math isn't just about following rules; it's about understanding the connections between different ideas and using them creatively.
Solving for x²
Okay, so let's get our hands dirty with some algebra! To effectively tackle this problem, our first major step is to isolate x² in the given equation, 6x² - 1 = hola. Remember, "hola" is still a placeholder for a number that we'll need to determine along the way. For now, let's treat it as a constant. To isolate x², we'll start by adding 1 to both sides of the equation. This gives us 6x² = hola + 1. Simple enough, right? Now, to get x² all by itself, we'll divide both sides of the equation by 6. This leaves us with x² = (hola + 1) / 6. Great! We've successfully isolated x². This is a significant milestone because it allows us to express x² in terms of "hola," which we can then use to find the value of our target expression, 36x² + 1/x². But hold on, we're not quite there yet. We still need to figure out what "hola" actually is. To do that, we'll need to think about how the value of x² affects the expression we're trying to find. This is where the puzzle pieces start to come together, and we'll see how everything connects. Remember, the goal is not just to find a solution but to understand the process and the reasoning behind each step. That's what makes math truly rewarding!