What is the condition for one root of the quadratic equation $a x^{2} + b x + c = 0$ to be twice the other?

b^{2} = 4 a c

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2 b^{2} = 9 a c

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c^{2} = 4 a + b^{2}

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c^{2} = 9 a - b^{2}

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Solution

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

$\Rightarrow 3 α = - \frac{b}{a} \Rightarrow α = - \frac{b}{3 a}$

and $2 α^{2} = \frac{c}{a} \Rightarrow 2 {[\frac{- b}{3 a}]}^{2} = \frac{c}{a}$

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

Hence, the required condition is $2 b^{2} = 9 a c$

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Similar questions

Q. What are the roots of the quadratic equation

a x^{2} + b x + c = 0 ?

Q. The condition that one root of the equation

a x^{2} + b x + c = 0

may be double the other, is:

Q. What is the nature of the roots of the quadratic equation

a x^{2} + b x + c = 0

if
(a)

b^{2} - 4 a c = 0

(b)

b^{2} - 4 a c < 0

The roots of a quadratic equation $a x^{2} + b x + c = 0$ are given by $\frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ , provided $b^{2} - 4 a c \geq 0$ .

Q. Which of the given formulae is the correct one to find out the roots of the quadratic equation

a x^{2} + b x + c = 0

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What is the condition for one root of the quadratic equation ax2+bx+c=0 to be twice the other?

The correct option is B 2b2=9acLet the roots of the quadratic equation be α and β.Here, α+2α=−ba and α(2α)=ca⇒3α=−ba⇒α=−b3aand 2α2=ca⇒2[−b3a]2=ca⇒2b29a2=ca⇒2b2=9acHence, the required condition is 2b2=9ac

What is the condition for one root of the quadratic equation $a x^{2} + b x + c = 0$ to be twice the other?