Solving 5p^4 - 405 = 0 Exact Form Solutions

In this comprehensive guide, we will delve into the process of solving the equation 5p^4 - 405 = 0, providing a detailed, step-by-step explanation to ensure clarity and understanding. We will explore the algebraic techniques required to isolate the variable p and determine its exact values. This equation, a quartic equation due to the presence of the p^4 term, presents an interesting challenge that we will tackle systematically. The goal is not just to find the solutions, but also to understand the underlying mathematical principles that govern them. Whether you are a student seeking to enhance your algebra skills or simply someone with an interest in mathematical problem-solving, this guide aims to provide valuable insights and a clear methodology for approaching similar equations.

1. Understanding the Equation: A Quartic in Disguise

Before diving into the solution, let's take a moment to understand the nature of the equation we're dealing with. The equation 5p^4 - 405 = 0 is a quartic equation, which means it's a polynomial equation of the fourth degree. However, it lacks the p^3, p^2, and p terms, making it a special case that can be solved using techniques similar to those used for quadratic equations. Recognizing this structure is crucial for choosing the most efficient solution method. In essence, we can treat p^4 as a squared term, which allows us to simplify the equation and apply familiar algebraic manipulations. This initial assessment sets the stage for a smoother solution process, highlighting the importance of observation and pattern recognition in mathematical problem-solving. Our strategy will involve isolating the p^4 term and then taking the fourth root to find the values of p. This approach underscores the power of algebraic manipulation in transforming complex equations into more manageable forms.

2. Isolating the p^4 Term: The First Step to the Solution

The first step in solving the equation 5p^4 - 405 = 0 is to isolate the p^4 term. This involves performing basic algebraic operations to get the term with the variable on one side of the equation and the constant term on the other side. To do this, we will add 405 to both sides of the equation. This maintains the equality and moves the constant term to the right side, effectively separating the variable term. The result of this operation is a simplified equation that brings us closer to our goal of solving for p. This step is fundamental in solving any algebraic equation, as it sets the stage for further manipulations and ultimately leads to the isolation of the variable. By isolating p^4, we create a clearer path towards finding its value and subsequently the values of p that satisfy the original equation. This process highlights the importance of methodical steps in solving mathematical problems, where each step builds upon the previous one to achieve the desired outcome.

5p^4 - 405 = 0
5p^4 = 405

Next, we need to divide both sides of the equation by 5 to completely isolate p^4. This operation further simplifies the equation and brings us closer to finding the value of p. Dividing by the coefficient of p^4 ensures that we have p^4 standing alone on one side of the equation, which is a crucial step before taking the fourth root. This step demonstrates the principle of maintaining balance in equations, where any operation performed on one side must also be performed on the other to preserve equality. The result of this division gives us a much simpler equation, making it easier to determine the value(s) of p that satisfy the original equation. This methodical approach of isolating the variable term is a cornerstone of algebraic problem-solving.

p^4 = 405 / 5
p^4 = 81

3. Taking the Fourth Root: Unveiling the Values of p

Now that we have isolated p^4 in the equation p^4 = 81, the next step is to take the fourth root of both sides. This operation will undo the exponentiation and allow us to solve for p. It's important to remember that when taking an even root, we need to consider both the positive and negative solutions, as both values raised to the fourth power will result in a positive number. In this case, we are looking for the numbers that, when raised to the fourth power, equal 81. The fourth root of 81 is 3, but we must also consider -3 as a possible solution since (-3)^4 also equals 81. This step highlights the importance of understanding the properties of exponents and roots, particularly when dealing with even powers. The inclusion of both positive and negative roots is a critical aspect of finding all possible solutions to the equation.

p = ±√[4]{81}

To find the fourth root of 81, we need to determine what number, when raised to the fourth power, equals 81. We know that 3 * 3 * 3 * 3 = 81, so the positive fourth root of 81 is 3. As mentioned earlier, we also need to consider the negative root, which is -3. Therefore, the solutions for p are 3 and -3. These are the real solutions to the equation. However, when dealing with even powers, there are also complex solutions to consider. Complex numbers extend the number system beyond real numbers and include the imaginary unit i, where i^2 = -1. To find the complex solutions, we need to delve into the realm of complex numbers and their properties. This exploration expands our understanding of the solutions to polynomial equations and introduces the concept of complex roots, which are fundamental in advanced algebra and calculus.

4. Finding the Complex Solutions: Expanding Our Solution Set

While we have found the real solutions to the equation p^4 = 81, there are also complex solutions to consider. Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as √(-1). To find the complex solutions, we can rewrite 81 in its polar form. Polar form represents a complex number in terms of its magnitude (distance from the origin) and its angle (in radians) from the positive real axis. This representation is particularly useful when dealing with roots of complex numbers. The polar form allows us to apply De Moivre's theorem, which provides a method for finding the roots of complex numbers. By expressing 81 in polar form, we can systematically find all four fourth roots, including the complex ones. This step demonstrates the power of complex numbers in providing a complete solution set for polynomial equations.

81 can be written as 81(cos(0) + i sin(0)) in polar form. To find the fourth roots, we use De Moivre's Theorem, which states that the nth roots of a complex number r(cos(θ) + i sin(θ)) are given by:

√[n]{r} [cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)]

where k = 0, 1, 2, ..., n-1.

In our case, r = 81, θ = 0, and n = 4. So, we have:

3 [cos((2πk)/4) + i sin((2πk)/4)]

for k = 0, 1, 2, 3.

Let's calculate the roots for each value of k:

• For k = 0:

  3 [cos(0) + i sin(0)] = 3

• For k = 1:

  3 [cos(π/2) + i sin(π/2)] = 3i

• For k = 2:

  3 [cos(π) + i sin(π)] = -3

• For k = 3:

  3 [cos(3π/2) + i sin(3π/2)] = -3i

Thus, the four solutions are 3, 3i, -3, and -3i. These solutions encompass both the real and complex roots of the equation, providing a complete understanding of the solution set. The complex solutions, 3i and -3i, demonstrate that the equation has roots beyond the real number line. This comprehensive approach to finding solutions highlights the importance of considering complex numbers when dealing with polynomial equations, especially those with even powers.

5. Exact Form of the Solutions: Presenting the Final Answer

Having determined all four solutions to the equation 5p^4 - 405 = 0, it's crucial to present them in their exact form. This means expressing the solutions without any approximations or decimal representations. The solutions we found are 3, -3, 3i, and -3i. These values represent the complete set of roots for the given quartic equation. The real solutions, 3 and -3, are integers, while the complex solutions, 3i and -3i, involve the imaginary unit i. Presenting the solutions in their exact form ensures mathematical precision and avoids any potential errors that might arise from rounding or approximation. This final step underscores the importance of clarity and accuracy in mathematical communication. By stating the solutions in their exact form, we provide a definitive and unambiguous answer to the problem.

Therefore, the solutions in exact form are:

• p = 3
• p = -3
• p = 3i
• p = -3i

These four solutions represent the complete solution set for the equation 5p^4 - 405 = 0. They include both the real and complex roots, providing a comprehensive understanding of the equation's behavior. This final presentation of the solutions in their exact form demonstrates the culmination of the problem-solving process, highlighting the importance of accuracy and clarity in mathematical communication.

Conclusion: Mastering Quartic Equations

In conclusion, we have successfully solved the quartic equation 5p^4 - 405 = 0, demonstrating a systematic approach to finding both real and complex solutions. The process involved isolating the p^4 term, taking the fourth root, and considering both positive and negative roots, as well as complex roots. By understanding the properties of exponents, roots, and complex numbers, we were able to determine the exact solutions: 3, -3, 3i, and -3i. This exercise provides valuable insights into solving higher-degree polynomial equations and reinforces the importance of algebraic manipulation, complex number theory, and attention to detail. Mastering these techniques enhances mathematical problem-solving skills and provides a solid foundation for tackling more complex equations in the future. The ability to solve such equations is crucial in various fields, including engineering, physics, and computer science, where mathematical models often involve polynomial equations. This comprehensive guide serves as a valuable resource for students and anyone interested in expanding their mathematical knowledge and skills.