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(2x/x-5)^2+10x/x-5-24=0

2 min read Jun 16, 2024
(2x/x-5)^2+10x/x-5-24=0

Solving the Equation: (2x/x-5)^2 + 10x/x-5 - 24 = 0

This equation appears complex, but we can solve it by utilizing a simple substitution and factoring. Let's break down the steps:

1. Substitution:

  • Let y = 2x / (x-5).

This simplifies the equation significantly:

y² + 10y - 24 = 0

2. Factoring:

  • Now we have a quadratic equation in terms of 'y'. We can factor this:

(y + 12)(y - 2) = 0

3. Solving for 'y':

  • This gives us two possible solutions for 'y':
  • y = -12
  • y = 2

4. Substituting Back:

  • Now we need to substitute back the original expression for 'y' and solve for 'x':

  • Case 1: y = -12

    • -12 = 2x / (x-5)
    • -12x + 60 = 2x
    • -14x = -60
    • x = 60/14 = 30/7

  • Case 2: y = 2

    • 2 = 2x / (x-5)
    • 2x - 10 = 2x
    • -10 = 0
    • This case leads to a contradiction, so there is no solution for x.

Solution:

Therefore, the only solution to the original equation is x = 30/7.

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