= Solution
Put $X=\omega^2$ and $S=\omega_1^2+\omega_2^2+4\Omega^2$. Its equation is $X^2-SX+\omega_1^2\omega_2^2=0$. If both effective stiffnesses are positive, the sum and product of roots are positive and $S^2-4\omega_1^2\omega_2^2\geq0$, so both roots are positive. They can coincide in a degenerate zero-rotation isotropic case; “two” need not mean distinct.
If $\omega_1^2<0<\omega_2^2$, the product is negative, so one root is negative. Its imaginary <frequency> gives a growing exponential displacement: <Centrifugal force>[centrifugal] softening has overwhelmed one elastic restoring direction, making the central configuration unstable. <Coriolis force>[Coriolis] coupling mixes the two coordinates but does not remove this negative-product instability.
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