# 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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