(4x-7)^2(2x-5)(x-1)=-1

4 min read Jun 16, 2024

Solving the Equation: (4x-7)^2(2x-5)(x-1) = -1

This problem presents a fourth-degree polynomial equation, which can be challenging to solve directly. However, we can leverage a few key insights and techniques to find the solutions.

1. Understanding the Equation

Let's break down the equation:

(4x-7)^2: This is a squared term, meaning it will always be non-negative (greater than or equal to zero).
(2x-5): This is a linear term.
(x-1): This is another linear term.

The product of these terms equals -1. Since the squared term is always non-negative, the remaining terms must have opposite signs to produce a negative product.

2. Identifying Potential Solutions

We can start by considering the following:

If (2x-5) > 0: Then (x-1) must be negative to make the product negative.
If (2x-5) < 0: Then (x-1) must be positive to make the product negative.

3. Solving for x

Let's explore the two possibilities:

Case 1: (2x-5) > 0 and (x-1) < 0

Solving (2x-5) > 0: x > 2.5
Solving (x-1) < 0: x < 1

These two conditions contradict each other, meaning there are no solutions in this case.

Case 2: (2x-5) < 0 and (x-1) > 0

Solving (2x-5) < 0: x < 2.5
Solving (x-1) > 0: x > 1

This case gives us a potential range of solutions: 1 < x < 2.5

Finding Exact Solutions

Since the equation is fourth degree, there might be up to four solutions. To find them, we could expand the entire expression, set it equal to -1, and use a method like the rational root theorem or numerical methods to find the roots. However, this process can be quite complex.

Alternative Approach: Numerical Methods

A more practical approach would be to use numerical methods, such as graphing calculators or software, to find the approximate solutions for x within the range 1 < x < 2.5. These tools can effectively identify the points where the graph of the function intersects the line y = -1.

4. Conclusion

While finding exact solutions for this equation can be challenging, by analyzing the properties of the equation and exploring potential ranges, we can narrow down the possible solutions. Using numerical methods provides a practical way to approximate the roots of this complex equation.