Progressive equations

Algebra Level 3

Consider the quadratic equation \[{ ax }^{ 2 }+bx+c=0\] with roots \(\alpha\) and \(\beta\), and whose coefficients \(a,b,c\) are distinct, non-zero real numbers in arithmetic progression.

If \(\frac { 1 }{ \alpha } +\frac { 1 }{ \beta } ,\ \alpha +\beta, \ { \alpha }^{ 2 }+{ \beta }^{ 2 }\) form a geometric progression, find \(\frac { a }{ c }.\)

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