(x-2)^2/2+18/(x-2)^2=7((x-2)/2-3/(x-2))+10

4 min read Jun 17, 2024

Solving the Equation: (x-2)^2/2 + 18/(x-2)^2 = 7((x-2)/2 - 3/(x-2)) + 10

This equation presents a challenge due to the presence of rational expressions and squares. Here's a step-by-step approach to solve it:

1. Simplify the Equation

Expand the expressions:
- (x-2)^2/2 = (x^2 - 4x + 4)/2
- 7((x-2)/2 - 3/(x-2)) = 7/2(x-2) - 21/(x-2)
Rewrite the equation: (x^2 - 4x + 4)/2 + 18/(x-2)^2 = 7/2(x-2) - 21/(x-2) + 10

2. Find a Common Denominator

Identify the LCD: The least common denominator for all fractions is 2(x-2)^2.
Multiply each term by the appropriate factor to achieve the LCD:
- (x^2 - 4x + 4)/2 * (x-2)^2/(x-2)^2 + 18/(x-2)^2 * 2/2 = 7/2(x-2) * (x-2)^2/(x-2)^2 - 21/(x-2) * 2(x-2)/2(x-2) + 10 * 2(x-2)^2/2(x-2)^2

3. Simplify and Rearrange

Expand and combine terms:
- (x^4 - 8x^3 + 20x^2 - 16x + 8)/2(x-2)^2 + 36/2(x-2)^2 = (7x^3 - 28x^2 + 28x - 56)/2(x-2)^2 - 42(x-2)/2(x-2)^2 + 20(x-2)^2/2(x-2)^2
Combine like terms:
- x^4 - 8x^3 + 20x^2 - 16x + 44 = 7x^3 - 28x^2 + 28x - 56 - 42x + 84 + 20x^2 - 80x + 80
Move all terms to one side:
- x^4 - 15x^3 + 48x^2 - 124x + 90 = 0

4. Solve the Quartic Equation

This is a quartic equation, and solving it directly can be complex. Here are some options:

Factoring (if possible): Try to factor the quartic equation. However, factoring quartics can be challenging.
Numerical Methods: Use numerical methods like the Newton-Raphson method or graphing calculators to find approximate solutions.
Software Solutions: Utilize mathematical software like Wolfram Alpha or MATLAB to find the roots of the equation.

5. Finding Solutions

The exact solutions for this equation are difficult to obtain algebraically. However, using numerical methods, we can find approximate solutions. There will likely be multiple solutions, and it's essential to check for any extraneous solutions that might arise due to squaring or multiplying by expressions involving variables.

Note: This equation does not have a simple, elegant solution. The steps above illustrate the process of simplifying and rearranging the equation to prepare it for further analysis using numerical methods or specialized software.