The condition that one root of $a x^{2} + b x + c = 0$ may be the double the other is:

b^{2} = 2 a c

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b^{2} = 3 a c

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2 b^{2} = 9 a c

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2 b^{2} = 3 a c

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Solution

The correct option is C $2 b^{2} = 9 a c$
Let $α$ and $β$ be the roots of the following equation.
It is given that $α = 2 β$
$α + β = 3 β = - \frac{b}{a}$ ...(i)
$α β = 2 β^{2} = \frac{c}{a}$ ...(ii)
Therefore $(\frac{b^{2}}{9 a^{2}}) = \frac{c}{2 a}$
$\Rightarrow 2 b^{2} = 9 a c$

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