Solving The Equation: 27p - 3 = 1 + 216/(9p-2) - 2 - 1 + 4p
Let's dive into solving this equation step-by-step. Equations like these, involving fractions and variables, might seem intimidating at first, but with a methodical approach, we can break them down and find the solution. So, grab your pencils, and let's get started!
Understanding the Equation
The equation we're tackling is: 27p - 3 = 1 + 216/(9p - 2) - 2 - 1 + 4p. First, let's simplify the right side of the equation to make it easier to work with. Combining the constants, 1 - 2 - 1 equals -2. So, the equation becomes: 27p - 3 = -2 + 216/(9p - 2) + 4p. This simplification already makes things a bit clearer.
Now, our goal is to isolate the variable 'p'. To do this, we need to get rid of the fraction. The fraction 216/(9p - 2) is the main obstacle here. To eliminate it, we can multiply both sides of the equation by (9p - 2). This will clear the fraction and give us a more manageable expression. Remember, whatever we do to one side of the equation, we must do to the other to maintain the balance. This is a fundamental principle in algebra. By multiplying both sides by (9p - 2), we ensure that the equality remains valid, and we can proceed with solving for 'p'. This step is crucial because it transforms the equation into a form that we can more easily manipulate and solve using standard algebraic techniques. Keep in mind that (9p - 2) cannot be equal to zero, as that would make the fraction undefined.
Clearing the Fraction
Okay, so we're multiplying both sides by (9p - 2). On the left side, we have (27p - 3) * (9p - 2). Expanding this gives us 243p^2 - 54p - 27p + 6, which simplifies to 243p^2 - 81p + 6. On the right side, we have (-2 + 216/(9p - 2) + 4p) * (9p - 2). Distributing (9p - 2) across each term, we get -2(9p - 2) + 216 + 4p(9p - 2). This simplifies to -18p + 4 + 216 + 36p^2 - 8p, which further simplifies to 36p^2 - 26p + 220. Now, our equation looks like this: 243p^2 - 81p + 6 = 36p^2 - 26p + 220. Notice how we've successfully eliminated the fraction, making the equation much easier to handle. This step is a game-changer!
Simplifying and Rearranging
Now that we've cleared the fraction, let's simplify and rearrange the equation to get all the terms on one side. We have 243p^2 - 81p + 6 = 36p^2 - 26p + 220. Subtracting 36p^2 from both sides gives us 207p^2 - 81p + 6 = -26p + 220. Adding 26p to both sides gives us 207p^2 - 55p + 6 = 220. Finally, subtracting 220 from both sides gives us 207p^2 - 55p - 214 = 0. So, now we have a quadratic equation in the standard form: 207p^2 - 55p - 214 = 0. This is a significant step because we can now use the quadratic formula or factoring to solve for 'p'. Quadratic equations might look scary, but they're just puzzles waiting to be solved!
Solving the Quadratic Equation
We've arrived at the quadratic equation: 207p^2 - 55p - 214 = 0. To solve this, we can use the quadratic formula: p = [-b ± sqrt(b^2 - 4ac)] / (2a). In our equation, a = 207, b = -55, and c = -214. Plugging these values into the quadratic formula, we get: p = [55 ± sqrt((-55)^2 - 4 * 207 * (-214))] / (2 * 207). Simplifying further, we have: p = [55 ± sqrt(3025 + 177192)] / 414. This becomes: p = [55 ± sqrt(180217)] / 414. Now, we need to find the square root of 180217, which is approximately 424.52. So, p = [55 ± 424.52] / 414. This gives us two possible solutions for 'p'. The first solution is p = (55 + 424.52) / 414, which is approximately 479.52 / 414, resulting in p ≈ 1.158. The second solution is p = (55 - 424.52) / 414, which is approximately -369.52 / 414, resulting in p ≈ -0.892. Therefore, the two possible values for 'p' are approximately 1.158 and -0.892. Quadratic formula to the rescue!
Checking the Solutions
Now that we have two possible solutions for 'p', we need to check if they are valid by plugging them back into the original equation: 27p - 3 = 1 + 216/(9p - 2) - 2 - 1 + 4p. Let's start with p ≈ 1.158. Plugging this into the equation, we get: 27(1.158) - 3 ≈ 1 + 216/(9(1.158) - 2) - 2 - 1 + 4(1.158). This simplifies to 31.266 - 3 ≈ 1 + 216/(10.422 - 2) - 2 - 1 + 4.632, which further simplifies to 28.266 ≈ 1 + 216/8.422 - 3 + 4.632. Continuing, we have 28.266 ≈ 1 + 25.645 - 3 + 4.632, which gives us 28.266 ≈ 28.277. This is very close, so p ≈ 1.158 is a valid solution.
Now let's check p ≈ -0.892. Plugging this into the equation, we get: 27(-0.892) - 3 ≈ 1 + 216/(9(-0.892) - 2) - 2 - 1 + 4(-0.892). This simplifies to -24.084 - 3 ≈ 1 + 216/(-8.028 - 2) - 2 - 1 - 3.568, which further simplifies to -27.084 ≈ 1 + 216/(-10.028) - 3 - 3.568. Continuing, we have -27.084 ≈ 1 - 21.54 - 3 - 3.568, which gives us -27.084 ≈ -27.108. This is also very close, so p ≈ -0.892 is also a valid solution. Both solutions check out! Always double-check your answers, guys!
Final Answer
After navigating through the maze of algebra, simplifying, rearranging, and applying the quadratic formula, we've successfully found the solutions to the equation 27p - 3 = 1 + 216/(9p - 2) - 2 - 1 + 4p. The solutions are approximately p ≈ 1.158 and p ≈ -0.892. We've also verified these solutions by plugging them back into the original equation to ensure they hold true. Equations like these can be challenging, but with a step-by-step approach and a solid understanding of algebraic principles, they can be conquered. So, next time you encounter a complex equation, remember to break it down, simplify, and tackle it one step at a time. You got this!
