If $\tan \alpha $ and $\tan \beta$ are the roots of the equation $ax^2+bx+c=0$ then the value of $\tan ( \alpha +\beta)$ is :

Using the sum and product of roots for the equation ax^2 + bx + c = 0, we have tan alpha + tan beta = -b/a and tan alpha tan beta = c/a. Substituting these values into the tangent addition formula, tan (alpha + beta) = (tan alpha + tan beta) / (1 - tan alpha tan beta), gives ((-b)/a) / (1 - c/a). Simplifying the denominator to (a - c)/a and dividing the terms results in b/(c - a).

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