Which of the following are the roots of the equation (A²/x) + B²/(x - 1) = 1, where A and B are real numbers, not equal to zero?

Multiply the entire equation A^2/x + B^2/(x - 1) = 1 by the common denominator x(x - 1) to clear the fractions. This yields the quadratic equation A^2(x - 1) + B^2(x) = x^2 - x. Expanding and moving all terms to one side gives x^2 - x - A^2(x) + A^2 - B^2(x) = 0, which simplifies to x^2 - (1 + A^2 + B^2)x + A^2 = 0. The product of the roots is A^2 and the sum is (1 + A^2 + B^2), meaning neither root can be 0 (since A is not 0) and neither root can be 1 (since the sum would require B^2 = 0, but B is not 0). Therefore, the roots can be any real number other than 0 and 1.

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