A quadratic equation is chosen from the set of all quadratic equations which are unchanged by squaring their roots. The chance that the chosen equation has equal roots is

For a quadratic equation ax^2 + bx + c = 0 with roots alpha and beta, squaring the roots yields a new equation a^2x^2 - ab(b^2 - 2ac)x + c(b^2 - 2ac)^2 = 0. Equating coefficients with the original equation leads to three possible conditions, one of which (a = c) has a probability of yielding equal roots while the other two (b = 0 and b^2 = 2ac) do not. Since only one out of these three equally likely conditions results in equal roots, the required chance is 1/3.

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