if-x-1-x-2-5-then-find-the-value-of-x-x-6-1-x-8-1-

Question Number 159961 by abdullah_ff last updated on 23/Nov/21

if x + \frac{1}{x} = 2 \sqrt{5} then find the value of

\frac{x (x^{6} - 1)}{x^{8} - 1}

Answered by Rasheed.Sindhi last updated on 23/Nov/21

\underset{―}{A H ard & D ifficult A pproach!}

x + \frac{1}{x} = 2 \sqrt{5}; \frac{x (x^{6} - 1)}{x^{8} - 1} = ?

x^{2} - 2 \sqrt{5} x + 1 = 0

x = \frac{2 \sqrt{5} \pm \sqrt{20 - 4}}{2} = \sqrt{5} \pm 2

▸ x \underset{\underset{―}{+ 2}}{- 1} = \sqrt{5} \underset{\underset{―}{+ 2}}{+ 1}, \sqrt{5} \underset{\underset{―}{+ 2}}{- 3} \dots \dots \dots (i)

x + 1 = \sqrt{5} + 3, \sqrt{5} - 1 \dots \dots \dots (i i)

(i) \times (i i) :

x^{2} - 1 = 8 + 4 \sqrt{5}, 8 - 4 \sqrt{5} \dots \dots . (i i i)

x^{2} + 1 = 10 + 4 \sqrt{5}, 10 - 4 \sqrt{5} \dots . . (i v)

(i i i) \times (i v) :

x^{4} - 1 = 160 + 72 \sqrt{5}, 160 - 72 \sqrt{5} \dots (v)

x^{4} + 1 = 162 + 72 \sqrt{5}, 162 - 72 \sqrt{5} \dots . (v i)

(v) \times (v i) :

x^{8} - 1 = 51840 \pm 23184 \sqrt{5}

▸ (i) : x - 1 = 1 + \sqrt{5}, - 3 + \sqrt{5} \dots \dots . . (v i i)

(i v) : x^{2} + 1 = 10 \pm 4 \sqrt{5}

x^{2} + x + 1 = (10 \pm 4 \sqrt{5}) + (\pm 2 + \sqrt{5})

= 12 + 5 \sqrt{5}, 8 - 3 \sqrt{5} \dots . (v i i i)

(v i i) \times (v i i i) :

x^{3} - 1 = 37 + 17 \sqrt{5}, - 39 + 17 \sqrt{5} \dots . . (i x)

x^{3} + 1 = 39 + 17 \sqrt{5}, - 37 + 17 \sqrt{5} \dots . (x)

(i x) \times (x) :

x^{6} - 1 = 2888 \pm 1292 \sqrt{5}

▸ \frac{x (x^{6} - 1)}{x^{8} - 1} = \frac{(\sqrt{5} \pm 2) (2888 \pm 1292 \sqrt{5})}{51840 \pm 23184 \sqrt{5}}

= \frac{\pm 12236 + 5472 \sqrt{5}}{51840 \pm 23184 \sqrt{5}} . \frac{51840 \mp 23184 \sqrt{5}}{51840 \mp 23184 \sqrt{5}}

= \frac{- 10944 \sqrt{5}}{{(51840)}^{2} - (23184^{2} . 5)}

= \frac{- 10944 \sqrt{5}}{- 103680}

= \frac{19 \sqrt{5}}{180}

Commented by abdullah_ff last updated on 25/Nov/21

Rasheed sir, you are r e a l l y great . .

Answered by Rasheed.Sindhi last updated on 23/Nov/21

▸ {(x + \frac{1}{x})}^{2} = {(2 \sqrt{5})}^{2} \Rightarrow x^{2} + \frac{1}{x^{2}} = 20 - 2

\Rightarrow {(x^{2} + \frac{1}{x^{2}})}^{2} = {(18)}^{2} \Rightarrow x^{4} + \frac{1}{x^{4}} = 322 \dots (i)

\Rightarrow {(x^{4} + \frac{1}{x^{4}})}^{2} = {(322)}^{2} \Rightarrow x^{8} + \frac{1}{x^{8}} = 322^{2} - 2 \dots (i i)

▸ {(x + \frac{1}{x})}^{3} = {(2 \sqrt{5})}^{3}

\Rightarrow x^{3} + \frac{1}{x^{3}} - 3 (x + \frac{1}{x}) = 8.5 . \sqrt{5}

\Rightarrow x^{3} + \frac{1}{x^{3}} - 3 (2 \sqrt{5}) = 8.5 . \sqrt{5}

\Rightarrow x^{3} + \frac{1}{x^{3}} = 46 \sqrt{5} \dots \dots . . (i i i)

(i) \times (i i i) :

(x^{4} + \frac{1}{x^{4}}) (x^{3} + \frac{1}{x^{3}}) = (322) (46 \sqrt{5})

x^{7} + \frac{1}{x^{7}} + x + \frac{1}{x} = (322) (46 \sqrt{5})

x^{7} + \frac{1}{x^{7}} + 2 \sqrt{5} = (322) (46 \sqrt{5})

x^{7} + \frac{1}{x^{7}} = (322) (46 \sqrt{5}) - 2 \sqrt{5}

▸ \frac{x (x^{6} - 1)}{x^{8} - 1} = \frac{x^{7}}{x^{8} - 1} - \frac{x}{x^{8} - 1}

= \frac{1}{\frac{x^{8} - 1}{x^{7}}} - \frac{1}{\frac{x^{8} - 1}{x}} = \frac{1}{x - \frac{1}{x^{7}}} - \frac{1}{x^{7} - \frac{1}{x}}

= \frac{(x^{7} - \frac{1}{x}) - (x - \frac{1}{x^{7}})}{(x - \frac{1}{x^{7}}) (x^{7} - \frac{1}{x})}

= \frac{(x^{7} + \frac{1}{x^{7}}) - (x + \frac{1}{x})}{(x^{8} + \frac{1}{x^{8}}) - 2}

= \frac{() - (x + \frac{1}{x})}{(x^{8} + \frac{1}{x^{8}}) - 2}

Commented by abdullah_ff last updated on 23/Nov/21

E^{X} C e l l e n t S i R

Answered by som(math1967) last updated on 23/Nov/21

\frac{x (x^{6} - 1)}{x^{8} - 1}

= \frac{\frac{x (x^{6} - 1)}{x^{4}}}{\frac{x^{8} - 1}{x^{4}}}

= \frac{x^{3} - \frac{1}{x^{3}}}{x^{4} - \frac{1}{x^{4}}}

= \frac{(x - \frac{1}{x}) (x^{2} + 1 + \frac{1}{x^{2}})}{(x - \frac{1}{x}) (x + \frac{1}{x}) (x^{2} + \frac{1}{x^{2}})} [(x - \frac{1}{x}) \neq 0]

= \frac{{(x + \frac{1}{x})}^{2} - 2 . x . \frac{1}{x} + 1}{2 \sqrt{5} {{(x + \frac{1}{x})}^{2} - 2 . x . \frac{1}{x}}}

= \frac{20 - 1}{2 \sqrt{5} (20 - 2)}

= \frac{19}{36 \sqrt{5}} = \frac{19 \sqrt{5}}{180}

Commented by abdullah_ff last updated on 23/Nov/21

this is great . thank you sir . .

Commented by Rasheed.Sindhi last updated on 23/Nov/21

E f f i c i e n t!

Answered by Tokugami last updated on 23/Nov/21

x + \frac{1}{x} = \frac{x^{2} + 1}{x} = 2 \sqrt{5}

{(x + \frac{1}{x})}^{2} = {(2 \sqrt{5})}^{2}

x^{2} + 2 + \frac{1}{x^{2}} = 20

\frac{1 + x^{4}}{x^{2}} = 18

\frac{x (x^{2} - 1) (x^{4} + x^{2} + 1)}{(x^{2} - 1) (x^{2} + 1) (x^{4} + 1)} = (\frac{x}{x^{2} + 1}) (\frac{x^{4} + 1 + x^{2}}{x^{4} + 1}) = (\frac{x}{x^{2} + 1}) (1 + \frac{x^{2}}{x^{4} + 1})

= (\frac{1}{2 \sqrt{5}}) (1 + \frac{1}{18})

= \frac{19}{36 \sqrt{5}}

Commented by abdullah_ff last updated on 23/Nov/21

t h a n k s for your k i n d help s i r

Commented by Rasheed.Sindhi last updated on 23/Nov/21

E fficient!