Number of values of $p$ for which the equation $(p^{2}-3p+2)x^{2}-(p^{2}-5p+4)x+p-p^{2}=0$ possess more then two roots is

A quadratic equation can possess more than two roots only if it reduces to the equation 0 = 0, meaning the coefficients of x squared, x, and the constant term must all be zero. Setting the coefficients to zero gives p squared minus 3p + 2 = 0, p squared minus 5p + 4 = 0, and p minus p squared = 0. The first two equations yield p = 1 as a common root, but the third equation requires p = 0 or p = 1, so p = 1 satisfies all three conditions simultaneously. Therefore, there is exactly one such value of p, giving 1.

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