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Question

What is the condition for one root of the quadratic equation $$ax^2 + bx + c = 0$$ to be twice the other?


A
b2=4ac
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B
2b2=9ac
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C
c2=4a+b2
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D
c2=9ab2
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Solution

The correct option is B $$2b^2=9ac$$
Let the roots of the quadratic equation be $$\alpha$$ and $$\beta$$.
Here, $$\alpha +2\alpha =-\dfrac{b}{a}$$ and $$\alpha (2\alpha) =\dfrac{c}{a}$$

$$\Rightarrow 3 \alpha  =-\dfrac{b}{a} \Rightarrow  \alpha =-\dfrac{b}{3a}$$

and $$ 2 \alpha^2  =\dfrac{c}{a} \Rightarrow 2 \left [ \dfrac{-b}{3a} \right ]^2 =\dfrac{c}{a}$$

$$\Rightarrow \dfrac{2b^2}{9a^2} =\dfrac{c}{a}\Rightarrow 2b^2=9ac$$

Hence, the required condition is $$2b^2=9ac$$

Maths

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