If the roots of the equation lx^2 + nx + n = | vedclass

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(A) Let the roots of the equation $lx^2 + nx + n = 0$ be $\alpha$ and $\beta$.Given that the ratio of the roots is $\frac{\alpha}{\beta} = \frac{p}{q}

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(A) Let the roots of the equation $lx^2 + nx + n = 0$ be $\alpha$ and $\beta$.Given that the ratio of the roots is $\frac{\alpha}{\beta} = \frac{p}{q}$.From the properties of quadratic equations,we have $\alpha + \beta = -\frac{n}{l}$ and $\alpha\beta = \frac{n}{l}$.We need to evaluate $\sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}} + \sqrt{\frac{n}{l}}$.Substituting $\frac{p}{q} = \frac{\alpha}{\beta}$ and $\frac{q}{p} = \frac{\beta}{\alpha}$,the expression becomes $\sqrt{\frac{\alpha}{\beta}} + \sqrt{\frac{\beta}{\alpha}} + \sqrt{\alpha\beta}$.This simplifies to $\frac{\sqrt{\alpha}}{\sqrt{\beta}} + \frac{\sqrt{\beta}}{\sqrt{\alpha}} + \sqrt{\alpha\beta} = \frac{\alpha + \beta}{\sqrt{\alpha\beta}} + \sqrt{\alpha\beta}$.Substituting the values $\alpha + \beta = -\frac{n}{l}$ and $\alpha\beta = \frac{n}{l}$,we get $\frac{-n/l}{\sqrt{n/l}} + \sqrt{n/l} = -\sqrt{\frac{n}{l}} + \sqrt{\frac{n}{l}} = 0$.

If the roots of the equation $lx^2 + nx + n = 0$ are in the ratio $p:q$,then $\sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}} + \sqrt{\frac{n}{l}} = $

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