The condition that one root of the equation a | vedclass

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(C) Let the roots of the quadratic equation $ax^2 + bx + c = 0$ be $\alpha$ and $3\alpha$.From the relationship between roots and coefficients:Sum of

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$3b^2 = 16ac$

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(C) Let the roots of the quadratic equation $ax^2 + bx + c = 0$ be $\alpha$ and $3\alpha$.From the relationship between roots and coefficients:Sum of roots: $\alpha + 3\alpha = -\frac{b}{a} \Rightarrow 4\alpha = -\frac{b}{a} \Rightarrow \alpha = -\frac{b}{4a}$.Product of roots: $\alpha \cdot 3\alpha = \frac{c}{a} \Rightarrow 3\alpha^2 = \frac{c}{a}$.Substitute the value of $\alpha$ into the product equation:$3\left(-\frac{b}{4a}\right)^2 = \frac{c}{a}$$3\left(\frac{b^2}{16a^2}\right) = \frac{c}{a}$$3b^2 = 16ac$.

The condition that one root of the equation $ax^2 + bx + c = 0$ is three times the other is

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