If the ratio of the roots of the equation ax^ | vedclass

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Question

(B) Let the roots of the equation $ax^2 + bx + c = 0$ be $p\alpha$ and $q\alpha$.According to the properties of roots:Sum of roots: $p\alpha + q\alpha

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$pqb^2 - (p + q)^2ac = 0$

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(B) Let the roots of the equation $ax^2 + bx + c = 0$ be $p\alpha$ and $q\alpha$.According to the properties of roots:Sum of roots: $p\alpha + q\alpha = -\frac{b}{a} \implies \alpha(p + q) = -\frac{b}{a} \implies \alpha = -\frac{b}{a(p + q)}$.Product of roots: $(p\alpha)(q\alpha) = \frac{c}{a} \implies pq\alpha^2 = \frac{c}{a}$.Substituting the value of $\alpha$ into the product equation:$pq \left( -\frac{b}{a(p + q)} \right)^2 = \frac{c}{a}$$pq \left( \frac{b^2}{a^2(p + q)^2} \right) = \frac{c}{a}$$pqb^2 = ac(p + q)^2$$pqb^2 - (p + q)^2ac = 0$.

If the ratio of the roots of the equation $ax^2 + bx + c = 0$ is $p:q$,then:

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