Hellenic Mathematical Olympiad ARCHIMEDES

Question

Determine the values of the parameter R for which the equation x^2 + ( - 2)x - ( - 1)(2 - 3) = 0 has two roots such that the one of them is equal to the square of the other.

Accepted Answer

1. We have the discriminant D = ( - 2)^2 + 4( - 1)(2 - 3) = (3 - 4)^2, and so the equation has the roots x_1 = - 1, x_2 = -2 + 3. Therefore we seek the values of for which: align* x_1 = x_2^2 or x_2 = x_1^2 & - 1 = (-2 + 3)^2 or -2 + 3 = ( - 1)^2 & - 1 = 4^2 - 12 + 9 or -2 + 3 = ^2 - 2 + 1 align* 4^2 - 13 + 10 = 0 or ^2 = 2 = 2 or = 54 or = 2 or = -2.