Solving The Exponential Equation: Step-by-Step Guide

Hey guys! Ever stumbled upon an equation that looks like it's from another dimension? Well, today, we're diving deep into one of those – an exponential equation! Specifically, we're going to break down how to solve: 5 * 4^x - 5 * 4^(x+1) + 5 * 4^(x+2) = 480. Sounds intimidating? Don't worry, we'll take it one step at a time, making it super easy to understand. So, grab your math hats, and let's get started! We'll explore this equation piece by piece, ensuring you're not just getting the answer but truly understanding the how and why behind it. By the end, you'll be tackling similar problems with confidence. Remember, math isn't about memorizing formulas; it's about understanding the process. Let's embark on this mathematical journey together and unravel the mysteries of exponential equations!

Understanding the Problem

Before we jump into solving, let's make sure we're all on the same page. Our mission is to find the value of 'x' that makes the equation true. The equation we're dealing with is 5 * 4^x - 5 * 4^(x+1) + 5 * 4^(x+2) = 480. Notice how 'x' is up there in the exponent? That's what makes this an exponential equation. To solve this, we need to use some clever algebraic manipulations and the properties of exponents. Think of it like a puzzle – we need to rearrange the pieces until the solution becomes clear. And just like any good puzzle, understanding the individual pieces is key. We'll focus on breaking down each term, identifying common factors, and simplifying the equation to make it more manageable. This initial understanding sets the stage for the actual solving process, ensuring we're not just blindly following steps but making informed decisions along the way. Ready to dive deeper? Let's move on to the next step!

Step 1: Simplifying the Equation

The key to cracking this equation lies in simplification. Remember those exponent rules from math class? They're about to become our best friends! Let's break down each term: 5 * 4^x is already in a pretty simple form. But what about 5 * 4^(x+1)? We can rewrite this using the rule a^(m+n) = a^m * a^n. So, 5 * 4^(x+1) becomes 5 * 4^x * 4^1, which is the same as 5 * 4^x * 4. Similarly, 5 * 4^(x+2) can be rewritten as 5 * 4^x * 4^2, or 5 * 4^x * 16. Now, our equation looks like this: 5 * 4^x - 5 * 4^x * 4 + 5 * 4^x * 16 = 480. See how 5 * 4^x appears in every term on the left side? That's our golden ticket! We can factor it out, making the equation much cleaner. Factoring out 5 * 4^x gives us: 5 * 4^x * (1 - 4 + 16) = 480. Now, let's simplify the expression inside the parentheses. What's 1 - 4 + 16? It's 13! So, our equation is now: 5 * 4^x * 13 = 480. We've taken a big leap in simplifying the original equation. Feels good, right? Next, we'll isolate the exponential term to get closer to our solution.

Step 2: Isolating the Exponential Term

Okay, we've simplified our equation to 5 * 4^x * 13 = 480. Now, let's get that 4^x all by itself on one side of the equation. To do this, we'll perform some basic algebraic operations. First, we need to get rid of the 5 * 13 that's hanging out with our exponential term. What's 5 * 13? It's 65. So, we have 65 * 4^x = 480. To isolate 4^x, we'll divide both sides of the equation by 65. This gives us: 4^x = 480 / 65. Now, let's simplify that fraction. Both 480 and 65 are divisible by 5, so let's divide both by 5. 480 / 5 = 96 and 65 / 5 = 13. So, we have 4^x = 96 / 13. Wait a minute... Is this correct? Let's double-check our calculations. It seems like there might have been a slight miscalculation. Let's go back a step. We had 5 * 4^x * 13 = 480. Dividing both sides by 65 (which is 5 * 13) should give us 4^x = 480 / 65. Upon closer inspection, 480 divided by 65 is not a clean integer. This indicates we need to re-evaluate our simplification process to ensure accuracy. Let's rewind and carefully check each step to pinpoint where we might have veered off course. Accuracy is paramount in math, and a small oversight can lead to a significant deviation in the result. Let's go back and make sure every step is as solid as it can be.

Step 3: Re-evaluating and Correcting the Approach

Alright, guys, it's crucial in problem-solving to recognize when something isn't quite right. We've hit a snag with 4^x = 480 / 65, as it doesn't lead to a straightforward solution. This is a good time to pause and double-check our steps. Going back to our factored equation: 5 * 4^x * (1 - 4 + 16) = 480, we correctly simplified the parentheses to get 5 * 4^x * 13 = 480. The next step was to isolate 4^x. We divided both sides by 5 * 13 = 65, leading to 4^x = 480 / 65. Here's where we need a closer look. While dividing is correct, the result 480 / 65 doesn't simplify neatly, suggesting a possible oversight in the problem's setup or a need for a different approach. Before resorting to more complex methods like logarithms, let's simplify the fraction 480/65 first. Dividing both numerator and denominator by 5 gives us 96/13. Now, we have 4^x = 96/13. This still doesn't look like a power of 4. Let's think about the original equation again: 5 * 4^x - 5 * 4^(x+1) + 5 * 4^(x+2) = 480. Maybe there's a simpler way to approach this. We've correctly simplified and isolated the exponential term, but the resulting fraction isn't cooperating. This is a clue to re-examine our initial strategy. Sometimes, the path we start on isn't the most direct one, and that's perfectly okay. Let's take a step back and look for alternative routes. What if we tried substituting the given options directly into the equation? It might save us some algebraic headaches. Let's explore that next!

Step 4: Testing the Answer Choices

Sometimes, the most efficient way to solve a problem, especially in a multiple-choice scenario, is to test the given options. We have the equation 5 * 4^x - 5 * 4^(x+1) + 5 * 4^(x+2) = 480, and the options are: A) x = 2, B) x = 3, C) x = 4, D) x = 5. Let's start with option A) x = 2. Substituting x = 2 into the equation, we get: 5 * 4^2 - 5 * 4^(2+1) + 5 * 4^(2+2) = 5 * 16 - 5 * 64 + 5 * 256. Now, let's calculate: 5 * 16 = 80, 5 * 64 = 320, and 5 * 256 = 1280. So, the equation becomes: 80 - 320 + 1280. Calculating further: 80 - 320 = -240, and -240 + 1280 = 1040. This does not equal 480, so option A is incorrect. Let's move on to option B) x = 3. Substituting x = 3 into the equation, we get: 5 * 4^3 - 5 * 4^(3+1) + 5 * 4^(3+2) = 5 * 64 - 5 * 256 + 5 * 1024. Now, let's calculate: 5 * 64 = 320, 5 * 256 = 1280, and 5 * 1024 = 5120. So, the equation becomes: 320 - 1280 + 5120. Calculating further: 320 - 1280 = -960, and -960 + 5120 = 4160. This also does not equal 480, so option B is incorrect. This process of elimination is super helpful, right? It narrows down our choices and gives us a clearer path forward. Let's keep going with the remaining options!

Step 5: Continuing to Test Answer Choices

We've tested options A and B, and neither of them satisfied the equation 5 * 4^x - 5 * 4^(x+1) + 5 * 4^(x+2) = 480. Let's keep going with our strategy of testing the answer choices. Next up is option C) x = 4. Substituting x = 4 into the equation, we get: 5 * 4^4 - 5 * 4^(4+1) + 5 * 4^(4+2) = 5 * 256 - 5 * 1024 + 5 * 4096. Now, let's calculate: 5 * 256 = 1280, 5 * 1024 = 5120, and 5 * 4096 = 20480. So, the equation becomes: 1280 - 5120 + 20480. Calculating further: 1280 - 5120 = -3840, and -3840 + 20480 = 16640. This definitely does not equal 480, so option C is incorrect. We're down to the last option, D) x = 5. If the math gods are smiling upon us, this should be our answer. Let's substitute x = 5 into the equation: 5 * 4^5 - 5 * 4^(5+1) + 5 * 4^(5+2) = 5 * 1024 - 5 * 4096 + 5 * 16384. Now, let's calculate: 5 * 1024 = 5120, 5 * 4096 = 20480, and 5 * 16384 = 81920. So, the equation becomes: 5120 - 20480 + 81920. Calculating further: 5120 - 20480 = -15360, and -15360 + 81920 = 66560. Oops! It looks like there was another miscalculation somewhere, or perhaps none of the provided options are correct. Let's take one more step back and re-examine the initial equation simplification to make absolutely sure we didn't miss anything. It's like being a detective, guys; sometimes you have to revisit the scene of the crime to find the missing clue!

Step 6: Final Review and Correct Solution

Okay, team, let's put on our detective hats one last time! We've tested all the answer choices, and none of them seem to work. This tells us we need to go back to the very beginning and meticulously review our steps. It's like debugging code – sometimes the error is in a place you least expect! We started with the equation 5 * 4^x - 5 * 4^(x+1) + 5 * 4^(x+2) = 480. We correctly rewrote it as 5 * 4^x - 5 * 4^x * 4 + 5 * 4^x * 16 = 480. Factoring out 5 * 4^x, we got 5 * 4^x * (1 - 4 + 16) = 480. Simplifying the parentheses, we have 5 * 4^x * 13 = 480. Now, let's isolate 4^x by dividing both sides by 5 * 13 = 65: 4^x = 480 / 65. Simplifying the fraction 480 / 65 by dividing both numerator and denominator by 5, we get 4^x = 96 / 13. Aha! Here's where we need to pause and think. The fraction 96/13 doesn't simplify to a power of 4, which means there may be an error in the original problem statement or the answer choices. However, if we assume there was a typo and the equation should have a clean solution, let's re-examine our steps to see if we can find a more suitable answer. Given the structure of the equation, it's likely the intended solution involves a small integer value for x. Since testing the answer choices didn't work, and our algebraic simplification led to a non-integer exponent, let's consider the possibility of an error in the problem itself. If we were to make an educated guess based on the pattern of the equation, we might look for a value of x that makes the equation close to 480. Without a clear path forward due to the non-integer result, the most prudent approach is to acknowledge the discrepancy and, if this were an exam, perhaps flag the question for review or clarification. In a real-world scenario, this is a perfect example of when to double-check the original data or problem statement for potential errors. So, while we couldn't find a definitive answer from the given options, we've learned the importance of meticulous checking and the courage to question the problem itself!