Multiple choice

Find a quadratic equation whose product of roots $x_1$ and $x_2$ is equal to $4$ and satisfying the relation $\displaystyle \frac{x_1}{x_1 - 1} + \frac{x_2}{x_2 - 1} = 2$.

  1. $x^2 - 2x + 4 = 0$
  2. $x^2 - 3x + 4 = 0$
  3. $x^2 - 2x + 3 = 0$
  4. $x^2 - 3x + 2 = 0$
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A Correct answer
Explanation

Given x1*x2 = 4. The equation is x1/(x1-1) + x2/(x2-1) = 2. Simplifying: (x1(x2-1) + x2(x1-1)) / ((x1-1)(x2-1)) = 2. This leads to 2x1x2 - (x1+x2) / (x1x2 - (x1+x2) + 1) = 2. Substituting x1x2=4: (8 - (x1+x2)) / (5 - (x1+x2)) = 2. 8 - (x1+x2) = 10 - 2(x1+x2). So x1+x2 = 2. The quadratic is x^2 - 2x + 4 = 0.

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