Introduction

Mathematics is a powerful tool that helps us solve complex equations, making it easier to understand various real-world phenomena. This article focuses on solving complex equations, specifically the equation 4x · 41/x 18. We will explore a systematic approach to solving this equation, including algebraic manipulation and the use of derivatives to ensure accuracy.

Solving the Equation: A Systematic Approach

Let's start with the equation 4x · 41/x 18. Our goal is to find the value of x.

Step 1: Introduce a Substitution

To simplify the equation, we can introduce a substitution. Let y 4x. This means that 41/x y1/x2. Substituting these into the original equation gives us:

y · y1/x2 18

Step 2: Algebraic Manipulation

At first glance, this equation might seem difficult to solve directly. However, by trying specific values for x, we can find a solution. Let's try x 2:

u201c42 · 41/2 16 · 2 18u201d

This works, so x 2 is a solution.

Step 3: Confirming the Solution

To check if there are other solutions, we need to analyze the behavior of the function f(x) 4x · 41/x. Let's find the derivative of f(x) to understand its behavior:

f'(x) 4x ln(4) - 41/x · (1/x2) ln(4)

We can see that the term 41/x · (1/x2) ln(4) decreases as x increases, while 4x ln(4) increases much faster for positive x. Therefore, f(x) is an increasing function for x 0.

Since f(x) is increasing and we found f(2) 18, this indicates that x 2 is the only positive solution.

Additional Considerations

Let's consider a similar equation and explore its solutions using a different approach:

The expression can be written as:

x8 - 1 18x4 or f(x) x8 - 18x4 - 1 0.

Letting z x4 transforms the equation into:

f(z) z2 - 18z - 1 0

Solving this quadratic equation will give us the values of z x4 that satisfy the relationship. This equation will have two solutions, and since the discriminant is positive, each solution for z x4 is a real number. For each solution, we take the fourth root to find x.

Roots of the Polynomial

The roots of f(z) z2 - 18z - 1 0 are:

z1 9 - 4√5 and z2 9 4√5

These are both positive numbers. Therefore, we need to solve:

x4 - (9 - 4√5) 0 and x4 - (9 4√5) 0

Both of these equations can be easily factored using the identity a2 - b2 (a b)(a - b). The roots of these equations are:

x ±√(9 - 4√5)1/4 and x ±i√(9 - 4√5)1/4

Additionally, the roots of:

x4 - (9 4√5) 0 are x ±√(9 4√5)1/4 and x ±i√(9 4√5)1/4

In total, there will be 8 solutions to the equation: 4 real number solutions and 4 complex number solutions.

Conclusion

Solving complex equations requires a combination of algebraic manipulation and the use of derivatives to understand the behavior of functions. By following a systematic approach, we can accurately find solutions to these equations. Whether you are dealing with exponential expressions or polynomial equations, the methods discussed in this article can help you solve a wide range of mathematical problems.