If and (

Question

If $\alpha $ and $\beta (\alpha <\beta )$ are the roots of the equation $(-2+\sqrt{3})(|\sqrt{x}-3|)+(x-6\sqrt{x})+(9-2\sqrt{3})=0,x\geqslant 0$, then $\sqrt{\frac{\beta }{\alpha }}+\sqrt{\alpha \beta }$ is equal to :

Accepted Answer

$\begin{gathered} (-2+\sqrt{3})(|\sqrt{x}-3|)+(x-6\sqrt{x}+9)-2\sqrt{3}=0,x\ge 0 \ |\sqrt{x}-3|=t \ \Rightarrow (-2+\sqrt{3})t+{t}^{2}-2\sqrt{3}=0 \ \Rightarrow t(t-2)+\sqrt{3}(t-2)=0 \ \Rightarrow (t+\sqrt{3})(t-2)=0 \ t+\sqrt{3}
e 0\Rightarrow t=2 \ \Rightarrow |\sqrt{x}-3|=2 \ \sqrt{x}-3=\pm 2 \ \sqrt{x}=5,1 \ x=25,1 \ \Rightarrow \alpha =1,\beta =25 \ \Rightarrow \sqrt{\frac{\beta }{\alpha }}+\sqrt{\alpha \beta }=10 \ \end{gathered}$

Solutions · 1