Multiple choice

Solve the quadratic equation: $\displaystyle \frac{x-1}{x-2}-\frac{x-2}{x-3}=\frac{x-5}{x-6}-\frac{x-6}{x-7}$

  1. $\displaystyle \frac{9}{2}$
  2. $0$
  3. $\dfrac 14$
  4. $-2$
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A Correct answer
Explanation

Equation: (1 - 1/(x-2)) - (1 - 1/(x-3)) = (1 - 1/(x-6)) - (1 - 1/(x-7)). Simplifies to: 1/(x-3) - 1/(x-2) = 1/(x-7) - 1/(x-6). 1/((x-3)(x-2)) = 1/((x-7)(x-6)). (x-3)(x-2) = (x-7)(x-6). x^2 - 5x + 6 = x^2 - 13x + 42. 8x = 36. x = 36/8 = 9/2.

AI explanation

On both sides, combine the terms in each fraction to get a constant difference of 1 in the numerators and denominators: 1 + 1/(x-2) - (1 + 1/(x-3)) = 1 + 1/(x-6) - (1 + 1/(x-7)). This simplifies to 1/(x-2) - 1/(x-3) = 1/(x-6) - 1/(x-7). Taking the difference of reciprocals gives -1/((x-2)(x-3)) = -1/((x-6)(x-7)), so (x-2)(x-3) = (x-6)(x-7). Expanding both quadratics gives x^2 - 5x + 6 = x^2 - 13x + 42, which simplifies to 8x = 36. Solving for x gives x = 9/2.

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