Arithmetic Progresion + Quadratic Equation = Nice Problem

Algebra Level pending

Given that \(c\), \(b\) and \(a\) are consecutive terms in a arithmetic progression. If the equation \(ax^2+bx+c=0\) has \(c\) as solution and \(a,b,c \ne 0\). The value of the common difference can be written as \(\frac{\sqrt{\alpha}}{\beta}\). Find the value of \(\alpha+\beta\)

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