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I have the following problem:

I have a Variable a for which I know that -5<a<-2 is valid.

Furthermore, I have the following two equations:

EV1 = -0.5*b + 0.5*Sqrt[b^2 - 4*c]
EV2 = -0.5*b - 0.5*Sqrt[b^2 - 4*c]

with

b = -a + 3.4 + L2
c = (a + 0.2)*(3.2 + L2) - 0.25*(-9.5 - L1)^2

Based on those equations, I want to compute an L1>2 and an L2 such that EV1<-2 and EV2<-2 for all a in the defined interval -5<a<-2.

I have already tried

$Assumptions = -5 < a < -2
(*1*)
Simplify[Reduce[EV1 < -2 && EV2 < 2 && L1 >= 2, {L1, L2}]]
(*2*)
Simplify[FindInstance[EV1 < 1.7 && EV2 < 1.7 && L1 >= 2, {L1, L2}]]

Unfortunately, both didn't work.

Does anybody know a possible way to compute an instance for L1 and L2 fulfilling the requirements?

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If you use rational numbers in your variables and equations, Reduce works and shows you, that you conditions can't get fullfilled.

      b = Rationalize[-a + 3.4 + L2, 0]
      c = Rationalize[(a + 0.2)*(3.2 + L2) - 0.25*(-9.5 - L1)^2, 0]

      EV1[L1_, L2_] = -1/2*b + 1/2*Sqrt[b^2 - 4*c] // Simplify
      EV2[L1_, L2_] = -1/2*b - 1/2*Sqrt[b^2 - 4*c] // Simplify

      (red = 
      Reduce[EV1[L1, L2] < -2 && EV2[L1, L2] < -2 && 
      L1 >= 2 && -5 < a < -2, {L1, L2}])

      (*  False  *)
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