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

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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