If-x-5-2-6-then-x-1-x-

Question Number 133981 by bemath last updated on 26/Feb/21

If x = 5 + 2 \sqrt{6} then \frac{x - 1}{\sqrt{x}} = ?

Answered by Dwaipayan Shikari last updated on 26/Feb/21

x = 5 + 2 \sqrt{6} = {(\sqrt{3} + \sqrt{2})}^{2}

\sqrt{x} - \frac{1}{\sqrt{x}} = \sqrt{3} + \sqrt{2} - \sqrt{3} + \sqrt{2} = 2 \sqrt{2}

Commented by malwan last updated on 26/Feb/21

\frac{2 \sqrt{6}}{2} = \sqrt{6} = \sqrt{2} \times \sqrt{3}

s u c h t h a t {(\sqrt{2})}^{2} + {(\sqrt{3})}^{2} = 5

t h a n k y o u s i r s h i k a r i

a n d t h i s m e t h o d c a n b e u s e d

t o f i n d t h e s q u a r e r o o t o f

c o m p l e x n u m b e r s

v e r y s i m p l e

Commented by Dwaipayan Shikari last updated on 26/Feb/21

G e n e r a l w a y

\sqrt{a + \sqrt{b}} = \sqrt{x} + \sqrt{y}

a + \sqrt{b} = x + y + 2 \sqrt{x y}

a = x + y b = 4 x y

x - y = \sqrt{a^{2} - b}

x = \frac{a + \sqrt{a^{2} - b}}{2} y = \frac{a - \sqrt{a^{2} - b}}{2}

\sqrt{a + \sqrt{b}} = \sqrt{\frac{a + \sqrt{a^{2} - b}}{2}} + \sqrt{\frac{a - \sqrt{a^{2} - b}}{2}}

\sqrt{5 + 2 \sqrt{6}} = \sqrt{5 + \sqrt{24}} = \sqrt{\frac{5 + \sqrt{25 - 24}}{2}} + \sqrt{\frac{5 - \sqrt{25 - 24}}{2}}

= \sqrt{3} + \sqrt{2}

Answered by malwan last updated on 26/Feb/21

x = 5 + 2 \sqrt{6}

\Rightarrow \frac{x - 1}{\sqrt{x}} = \frac{4 + 2 \sqrt{6}}{\sqrt{5 + 2 \sqrt{6}}} \times \frac{\sqrt{5 + 2 \sqrt{6}}}{\sqrt{5 + 2 \sqrt{6}}}

= \frac{2 (2 + \sqrt{6}) \sqrt{5 + 2 \sqrt{6}}}{5 + 2 \sqrt{6}} \times \frac{5 - 2 \sqrt{6}}{5 - 2 \sqrt{6}}

= \frac{2 (10 + \sqrt{6} - 12) \sqrt{5 + 2 \sqrt{6}}}{25 - 4 \times 6}

= \frac{2 (\sqrt{6} - 2) \sqrt{5 + 2 \sqrt{6}}}{1}

= 2 \sqrt{(6 - 4 \sqrt{6} + 4) (5 + 2 \sqrt{6})}

= 2 \sqrt{2 (5 - 2 \sqrt{6}) (5 + 2 \sqrt{6})}

= 2 \sqrt{2 (25 - 24)} = 2 \sqrt{2 \times 1} = 2 \sqrt{2}

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