Question

If x=7+2√10 then √x+3/√x=

1) 0
2) 1
3) 2√5
4) 2√2

​

Answer 1

$\large \underline \bold{Given :-}$

$\: \: \: \: \:$$\small \bold{x = 7 + 2\sqrt{10}}$

$\large \underline \bold{To \: Find :-}$

$\: \: \: \:$$\small \bold{\sqrt{x} + \dfrac{3}{\sqrt{x}} = ?}$

$\large \underline \bold{Using \: Formula :-}$

$\small \bold{(a^{2} + b^{2} + 2ab) = (a + b)^{2}}$

$\large \underline \bold{Solution :-}$

$\small \bold{As \: we \: given -}$

$\: \: \: \: \:$$\small \bold{x = 7 + 2\sqrt{10}}$

$\small \bold{Now \: ,}$

$\: \: \: \: \: \: \: \: \: \:$$\small \bold{x + \dfrac{9}{x}}$

$\small \bold{=> (7 + 2\sqrt{10}) + \dfrac{9}{(7 + 2\sqrt{10})}}$

$\small \bold{=> \dfrac{(7 + 2\sqrt{10})^{2} + 9}{(7 + 2\sqrt{10})}}$

$\small \bold{=> \dfrac{(7)^{2} + (2\sqrt{10})^{2} + 2(7)(2\sqrt{10}) + 9}{(7 + 2\sqrt{10})}}$

$\small \bold{=> \dfrac{49 + 40 + 28\sqrt{10} + 9}{(7 + 2\sqrt{10})}}$

$\small \bold{=> \dfrac{98 + 28\sqrt{10}}{(7 + 2\sqrt{10})}}$

$\small \bold{=> \dfrac{14(7 + 2\sqrt{10})}{(7 + 2\sqrt{10})}}$

$\small \bold{=> 14}$

$\small \bold{So \: ,}$

$\: \: \: \: \: \: \: \: \: \: \: \:$$\small \bold{x + \dfrac{9}{x} = 14}$

$\: \: \: \:$$\small \bold{x + \dfrac{9}{x} + 6 = 14 + 6}$

$\: \: \: \:$$\small \bold{x + \dfrac{9}{x} + 6 = 20}$

$\small \bold{(\sqrt{x})^{2} + (\dfrac{3}{\sqrt{x}})^{2} + 2(\cancel{\sqrt{x}})(\dfrac{3}{\cancel{\sqrt{x}}}) = 20}$

$\:$$\small \bold{(\sqrt{x} + \dfrac{3}{\sqrt{x}})^{2} = 20}$

$\: \:$$\small \bold{(\sqrt{x} + \dfrac{3}{\sqrt{x}}) =\sqrt{20}}$

$\: \:$$\small \bold{(\sqrt{x} + \dfrac{3}{\sqrt{x}}) = 2\sqrt{5}}$