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

3 min read Jun 16, 2024
(x+1/x-4)2=x+8/x-2

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

This equation presents a challenge due to the presence of fractions and squares. Let's break down the steps to solve it:

1. Simplify the Left-Hand Side

  • Expand the square: (x+1/x-4)2 = (x+1/x-4) * (x+1/x-4)
  • Use FOIL (First, Outer, Inner, Last):
    • (x * x) + (x * 1/x-4) + (1/x-4 * x) + (1/x-4 * 1/x-4)
  • Simplify: x² + 1 + 1 + 1/(x-4)²

The left-hand side simplifies to x² + 2 + 1/(x-4)²

2. Manipulate the Equation

  • Bring all terms to one side: x² + 2 + 1/(x-4)² - (x+8/x-2) = 0
  • Find a Common Denominator: To combine the fractions, find the least common multiple of (x-4)² and (x-2): (x-4)² * (x-2)
  • Rewrite each term with the common denominator:
    • x² * (x-4)²(x-2) / (x-4)²(x-2)
    • 2 * (x-4)²(x-2) / (x-4)²(x-2)
    • 1 / (x-4)²
    • (x+8/x-2) * (x-4)²(x-2) / (x-4)²(x-2)

3. Solve the Equation

  • Simplify the equation: After simplifying all the terms and combining them, you will have a polynomial equation.
  • Solve the polynomial equation: This can involve factoring, using the quadratic formula, or other methods depending on the degree of the polynomial.

Important Notes:

  • Domain Restrictions: Be mindful of values of 'x' that would make any of the denominators equal to zero. These values are excluded from the solution set.
  • Extraneous Solutions: Always check your solutions by plugging them back into the original equation to ensure they are valid. Some solutions might arise during the simplification process but might not satisfy the original equation.

Solving this equation is a complex process that requires careful simplification and algebraic manipulation. By following the steps outlined above, you can systematically work towards finding the solutions.

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