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$$\alpha$$ and $$\beta$$ are the roots of the equation $$ax^2 + bx + c = 0$$. What is the condition for which $$\beta = 5 \alpha$$ ?
Since $$\beta\ =\ 5\alpha\ \ $$, we can say that the product of roots ($$5\alpha\ ^2\ \ $$) = $$\ \frac{\ c}{a}$$ ............ (1)
and sum of roots ($$6\alpha\ $$) = $$\ \frac{\ -b}{a}$$ ................. (2)
Hence, $$\alpha\ $$ = $$\ \frac{\ -b}{6a}$$ ............. (3)
Substituting (3) in (1), we get $$5\times\ \ \frac{\ b^2}{a^2}$$ = $$\ \frac{\ c}{a}$$
or, $$5b^2 = 36 ac$$
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