Let one root of the equation ax^{2} + bx + c | vedclass

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Question

(C) Given the quadratic equation $ax^{2} + bx + c = 0$ with roots $\alpha$ and $2\alpha$.From the properties of roots:Sum of roots $= \alpha + 2\alpha

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$2b^{2} = 9ac$

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(C) Given the quadratic equation $ax^{2} + bx + c = 0$ with roots $\alpha$ and $2\alpha$.From the properties of roots:Sum of roots $= \alpha + 2\alpha = 3\alpha = -\frac{b}{a} \implies \alpha = -\frac{b}{3a} \quad (1)$Product of roots $= \alpha \cdot 2\alpha = 2\alpha^{2} = \frac{c}{a} \implies \alpha^{2} = \frac{c}{2a} \quad (2)$Substituting $(1)$ into $(2)$:$\left(-\frac{b}{3a}\right)^{2} = \frac{c}{2a}$$\frac{b^{2}}{9a^{2}} = \frac{c}{2a}$$2b^{2} = 9a^{2} \cdot \frac{c}{a}$$2b^{2} = 9ac$

Let one root of the equation $ax^{2} + bx + c = 0$ be $\alpha$. Then the other root is $2\alpha$. Find the relation between $a, b,$ and $c$.

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