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Parametrized quadratic equation

Source: Czech and Slovak Olympiad 1964, National Round, Problem 3

June 22, 2024
parameterizationquadraticsquadratic polynomialalgebra

Problem Statement

Determine all values of parameter α∈[0,2π]\alpha\in [0,2\pi] such that the equation (2cos⁡α−1)x2+4x+4cos⁡α+2=0(2\cos\alpha-1)x^2+4x+4\cos\alpha+2=0 has 1) a positive root x1x_1, 2) if a second root x2x_2 exists and if x2≠x1x_2\neq x_1, the x2≤0x_2\leq 0.
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