(x-1)^2/8+8/(x-1)^2=7(x-1/4-2/x-1)-1

5 min read Jun 17, 2024

Solving the Equation: (x-1)^2/8 + 8/(x-1)^2 = 7(x-1/4 - 2/x-1) - 1

This equation looks intimidating at first glance, but with careful manipulation, we can simplify it and find its solutions. Let's break down the process step by step:

1. Simplifying the Right-hand Side

The right-hand side of the equation needs to be simplified before we can proceed. Let's focus on the terms inside the parentheses:

x - 1/4: This term can be rewritten with a common denominator as (4x - 1)/4.
-2/x-1: We can rewrite this term as -2/(x-1).

Now, the right-hand side becomes:

7( (4x - 1)/4 - 2/(x-1) ) - 1

Let's distribute the 7:

(7/4)(4x-1) - 14/(x-1) - 1

Simplifying further:

7x - 7/4 - 14/(x-1) - 1

2. Combining Terms and Making a Substitution

Now, we can rewrite the equation with the simplified right-hand side:

(x-1)^2/8 + 8/(x-1)^2 = 7x - 7/4 - 14/(x-1) - 1

Let's combine the constants on the right-hand side and make a substitution to simplify the equation:

(x-1)^2/8 + 8/(x-1)^2 = 7x - 11/4 - 14/(x-1)

Let's substitute y = (x-1):

y^2/8 + 8/y^2 = 7(y+1) - 11/4 - 14/y

3. Multiplying by the Common Denominator

To get rid of the fractions, let's multiply both sides of the equation by the common denominator, which is 8y^2:

y^4 + 64 = 56y^3 + 56y^2 - 22y^2 - 112y

4. Rearranging and Simplifying

Let's rearrange the terms to get a standard polynomial equation:

y^4 - 56y^3 + 34y^2 + 112y + 64 = 0

This is a fourth-degree polynomial equation, which can be challenging to solve directly. However, we can try to factor it. By observing the coefficients and constants, we can see that y = 4 is a root of the equation. This means that (y-4) is a factor of the polynomial.

We can use polynomial long division or synthetic division to divide the polynomial by (y-4) and obtain the other factor:

y^3 - 52y^2 + 18y + 16 = 0

Unfortunately, this cubic equation is still difficult to solve analytically. We might need to resort to numerical methods or graphing techniques to find approximate solutions for this part.

5. Back-Substituting and Finding Solutions

Once we find the roots for the cubic equation, we can back-substitute y = (x-1) to find the solutions for the original equation.

Keep in mind that the original equation has potential restrictions on the domain due to the presence of (x-1) in the denominator. We need to ensure that our solutions don't lead to division by zero.

In conclusion, solving the equation (x-1)^2/8 + 8/(x-1)^2 = 7(x-1/4 - 2/x-1) - 1 involves simplifying the equation, making substitutions, and ultimately solving a fourth-degree polynomial equation. While finding the exact solutions might require numerical methods, the steps outlined above provide a systematic approach to tackle this complex problem.