Solving 5p^4 - 405 = 0 Find The Solution Set
In this article, we will delve into the step-by-step process of solving the equation 5p^4 - 405 = 0. This problem falls under the category of algebraic equations, specifically a polynomial equation of degree four. Understanding how to solve such equations is a fundamental skill in mathematics, with applications spanning various fields like physics, engineering, and computer science. We will not only provide the solution but also elaborate on the underlying concepts and techniques, ensuring a clear and thorough understanding of the solution process. Before we dive into the specifics, it's crucial to recognize the structure of the equation. It's a quartic equation, meaning the highest power of the variable 'p' is 4. Solving quartic equations can sometimes be complex, but in this particular case, the equation has a simplified form that allows us to employ straightforward algebraic methods. The key to solving this equation lies in isolating the variable 'p' and then finding the roots, which are the values of 'p' that satisfy the equation. We'll use techniques such as factoring, applying the square root property, and understanding complex numbers to arrive at the complete solution set. Each step will be explained in detail to ensure clarity and comprehension. So, let's embark on this mathematical journey and unravel the solution to the equation 5p^4 - 405 = 0. This detailed explanation will not only equip you with the answer but also enhance your problem-solving skills in algebra.
Step 1: Isolate the Term with p^4
The first crucial step in solving the equation 5p^4 - 405 = 0 is to isolate the term containing the variable 'p' raised to the power of 4. This involves moving the constant term to the other side of the equation. To achieve this, we add 405 to both sides of the equation. This maintains the balance of the equation and helps us to progressively isolate the term we are interested in. Adding 405 to both sides gives us a new equation: 5p^4 = 405. Now, we have successfully moved the constant term to the right side, bringing us closer to isolating p^4. This is a common technique in solving algebraic equations, where we manipulate the equation to get the variable term by itself. The next step involves getting rid of the coefficient that is multiplying p^4, which in this case is 5. To do this, we will divide both sides of the equation by 5. This operation will further simplify the equation and bring us closer to the ultimate goal of finding the values of 'p' that satisfy the equation. Remember, the fundamental principle here is to perform the same operation on both sides of the equation to maintain equality. By isolating the term with p^4, we set the stage for the subsequent steps that will lead us to the solution set. This methodical approach is key to solving a wide range of algebraic problems. Understanding this initial step is paramount for anyone looking to master the art of equation solving.
Step 2: Divide Both Sides by 5
Following the isolation of the term 5p^4, the next logical step is to eliminate the coefficient multiplying p^4. In our equation, 5p^4 = 405, the coefficient is 5. To remove this coefficient, we perform the inverse operation, which is division. We divide both sides of the equation by 5. This maintains the equality of the equation while simplifying it further. When we divide both sides by 5, we get: (5p^4) / 5 = 405 / 5. This simplifies to p^4 = 81. Now, we have successfully isolated p^4 on one side of the equation. This is a significant milestone in solving the equation, as we have reduced it to a simpler form. The equation p^4 = 81 tells us that we are looking for a number which, when raised to the power of 4, equals 81. This opens up the possibility of finding both real and complex solutions. Before we proceed, it's important to understand that raising a number to the power of 4 means multiplying the number by itself four times. To find the value(s) of 'p', we need to find the fourth root of 81. This step involves understanding the concept of roots and how they relate to exponents. The fourth root of a number is a value that, when raised to the fourth power, gives the original number. Finding the fourth root will lead us to the solution set for the equation. This step is crucial and lays the groundwork for the final steps in finding the solution.
Step 3: Take the Fourth Root of Both Sides
Having isolated p^4 in the equation p^4 = 81, the next step is to find the value(s) of 'p'. This involves taking the fourth root of both sides of the equation. The fourth root of a number is the value that, when raised to the power of 4, equals that number. When dealing with even roots, like the fourth root, it's crucial to remember that there are both positive and negative real solutions, as well as complex solutions. Taking the fourth root of both sides of p^4 = 81 gives us p = ±√[4]{81}. Here, √[4]{81} represents the fourth root of 81. We use the '±' symbol to indicate that there are both positive and negative real solutions. Now, we need to determine the fourth root of 81. We know that 3 * 3 * 3 * 3 = 81, so the positive fourth root of 81 is 3. Therefore, one solution is p = 3. However, we must also consider the negative root. Since (-3) * (-3) * (-3) * (-3) = 81 as well, the negative fourth root of 81 is -3. Thus, another solution is p = -3. These are the real solutions to the equation. But, because we are dealing with a fourth-degree polynomial, we also need to consider the possibility of complex solutions. Complex solutions arise when we take the even root of a negative number or when we consider the complete set of roots in the complex plane. Understanding this step is essential because it highlights the importance of considering all possible solutions, both real and complex, when solving polynomial equations.
Step 4: Identify Real and Complex Solutions
After taking the fourth root in the equation p^4 = 81, we found that p = ±3. These are the real solutions, meaning they are real numbers that satisfy the equation. However, since we are dealing with a fourth-degree polynomial equation, we expect to find four solutions in total, counting both real and complex solutions. To find the complex solutions, we need to delve into the realm of complex numbers. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1 (i = √-1). To find the complex solutions, we can rewrite the equation p^4 = 81 as p^4 - 81 = 0. This can be factored as a difference of squares: (p^2 - 9)(p^2 + 9) = 0. The first factor, p^2 - 9 = 0, gives us the real solutions we already found, p = 3 and p = -3. The second factor, p^2 + 9 = 0, will lead us to the complex solutions. To solve p^2 + 9 = 0, we can rearrange it as p^2 = -9. Taking the square root of both sides, we get p = ±√-9. Since the square root of -1 is i, we can rewrite this as p = ±√(9 * -1) = ±√(9) * √(-1) = ±3i. Therefore, the complex solutions are p = 3i and p = -3i. Now, we have identified all four solutions to the equation: two real solutions (3 and -3) and two complex solutions (3i and -3i). This step is crucial in understanding the complete solution set of a polynomial equation, as it highlights the existence and importance of complex roots in addition to real roots. Understanding complex solutions is fundamental in many areas of mathematics and physics.
Step 5: Write the Solution Set
Having identified all the solutions to the equation 5p^4 - 405 = 0, the final step is to express the solution set. The solution set is a collection of all values of 'p' that satisfy the equation. We found two real solutions, p = 3 and p = -3, and two complex solutions, p = 3i and p = -3i. To write the solution set, we typically enclose these values within curly braces. The order in which the solutions are listed within the set does not matter. Therefore, the solution set for the equation 5p^4 - 405 = 0 is {-3, 3, -3i, 3i}. This set includes all the values of 'p' that, when substituted back into the original equation, will make the equation true. It's important to verify these solutions by substituting them back into the original equation to ensure they are correct. For example, if we substitute p = 3 into the equation, we get 5(3^4) - 405 = 5(81) - 405 = 405 - 405 = 0, which confirms that p = 3 is indeed a solution. Similarly, we can verify the other solutions. Writing the solution set is the culmination of the problem-solving process. It provides a clear and concise answer to the problem. Understanding how to express the solution set is as important as finding the solutions themselves. This final step completes our comprehensive guide to solving the equation 5p^4 - 405 = 0, showcasing the various algebraic techniques involved and the importance of considering both real and complex solutions.
Therefore, the solution set is { -3, 3, -3i, 3i }.
