How to get rid of a,b,c?

Solution:

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$a,b,c$ are in a Geometric Progression thus they can also be written the form of $\frac{b}{q} , b , b.q$ where $q$ is the common quotient.Hence follows :

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$\frac{b}{q} + b + b.q = \frac {b + bq + bq^2}{q} = xb$

$\rightarrow xq = 1+q+q^2 \rightarrow q^2 + (1-x)q +1 =0$

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Now we know that the numbers are $Real$ and $Distinct$ hence the we must have:

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$\Delta > 0 \rightarrow x^2 -2x -3 > 0 \rightarrow x<-1 \land x>3 \rightarrow \alpha^2 + \beta^2 = 10$