Multiple choice

The roots of the equation $\displaystyle \frac{x}{x-1}+\frac{x-1}{x}=2\frac{1}{2}$ are :

  1. $1, 2$
  2. $-2, -1$
  3. $-2, 1$
  4. $2, -1$
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D Correct answer
Explanation

Let x/(x-1) = y. Then y + 1/y = 5/2. Solving y^2 - 2.5y + 1 = 0 gives y = 2 or y = 0.5. If x/(x-1) = 2, x = 2x - 2, so x = 2. If x/(x-1) = 0.5, 2x = x - 1, so x = -1.

AI explanation

Multiply both sides by 2x(x-1) to clear the denominators: 2x^2 + 2(x-1)^2 = 5x(x-1). Expanding and simplifying gives 2x^2 + 2x^2 - 4x + 2 = 5x^2 - 5x, which rearranges to the quadratic equation x^2 - x - 2 = 0. Factoring this yields (x-2)(x+1) = 0, so the roots are 2 and -1.

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