Progressive equations

Algebra Level 3

Consider the quadratic equation ax2+bx+c=0{ ax }^{ 2 }+bx+c=0 with roots α\alpha and β\beta, and whose coefficients a,b,ca,b,c are distinct, non-zero real numbers in arithmetic progression.

If 1α+1β, α+β, α2+β2\frac { 1 }{ \alpha } +\frac { 1 }{ \beta } ,\ \alpha +\beta, \ { \alpha }^{ 2 }+{ \beta }^{ 2 } form a geometric progression, find ac.\frac { a }{ c }.

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