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If u and v are the roots of $$ax^{2} + bx + c = 0$$, then what is $$\sqrt{\frac{2\sqrt{ac}-b}{c}}$$?
Using the sum and product of roots formula, we getΒ $$\frac{c}{a}=uv$$, and hence,Β $$c=auv$$
AndΒ $$-\frac{b}{a}=u+v$$, and hence,Β $$-b=a\left(u+v\right)$$
Putting values of c and b in the equation we get,Β
$$\sqrt{\ \frac{2\sqrt{\ ac}-b}{c}}=\sqrt{\ \frac{2\sqrt{\ a\left(auv\right)}+a\left(u+v\right)}{auv}}$$
=$$\sqrt{\ \frac{2\sqrt{\ a^2uv}+a\left(u+v\right)}{uav}}=\sqrt{\ \frac{2\sqrt{\ uv}+\left(u+v\right)}{uv}}$$
=$$\sqrt{\ \frac{\left(\sqrt{\ u}+\sqrt{\ v}\right)^2}{uv}}=\sqrt{\ \frac{\left(\sqrt{\ u}+\sqrt{\ v}\right)^2}{\sqrt{\ uv}^2}}=\sqrt{\ \left(\frac{\sqrt{\ u}+\sqrt{\ v}}{\sqrt{\ uv}}\right)^2}$$
=$$\frac{1}{\sqrt{\ u}}+\frac{1}{\sqrt{\ v}}$$
Therefore, Option A is the correct answer.Β
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