if-x-3-1-x-3-52-find-the-value-of-x-4-1-x-4-

Question Number 119193 by mr W last updated on 22/Oct/20

i f x^{3} + \frac{1}{x^{3}} = 52, f i n d t h e v a l u e o f

x^{4} + \frac{1}{x^{4}} = ?

Commented by PRITHWISH SEN 2 last updated on 22/Oct/20

x + \frac{1}{x} = 4, - 2 + 3 i, - 2 - 3 i

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

∴ x^{4} + \frac{1}{x^{4}} = 194 or - 129 + 168 i or - 105 - 168 i

Commented by mr W last updated on 22/Oct/20

t h a n k s!

Commented by PRITHWISH SEN 2 last updated on 22/Oct/20

welcome

Answered by TANMAY PANACEA last updated on 22/Oct/20

{(x + \frac{1}{x})}^{3} - 3 (x + \frac{1}{x}) = 52

a^{3} - 3 a - 52

= a^{3} - 3 a - (64 - 12)

= a^{3} - 64 - 3 (a - 4)

= (a - 4) (a^{2} + 4 a + 16) - 3 (a - 4)

= (a - 4) (a^{2} + 4 a + 13)

= (a - 4) {{(a + 2)}^{2} + 9}

s o a = 4 \to x + \frac{1}{x} = 4

x^{2} + \frac{1}{x^{2}} = {(x + \frac{1}{x})}^{2} - 2 = 14

x^{4} + \frac{1}{x^{4}} = {(x^{2} + \frac{1}{x^{2}})}^{2} - 2 = 196 - 2 = 194

Commented by mr W last updated on 22/Oct/20

c o r r e c t, t h a n k s!

Commented by TANMAY PANACEA last updated on 22/Oct/20

s i r i h a v e n o t c o n s i d e r e d c o m l e x s o l u t i o n

Answered by 1549442205PVT last updated on 23/Oct/20

x^{3} + \frac{1}{x^{3}} = 52 \Leftrightarrow x^{6} - {52 x}^{3} + 1 = 0

Δ^{'} = 26^{2} - 1 = 675 = {(15 \sqrt{3})}^{2}

\Rightarrow x^{3} = 26 \pm 15 \sqrt{3} = {(2 \pm \sqrt{3})}^{3} \Rightarrow x = 2 \pm \sqrt{3}

Since (2 + \sqrt{3}) (2 - \sqrt{3}) = 1 \Rightarrow x + \frac{1}{x}

= 2 + \sqrt{3} + 2 - \sqrt{3} = 4

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

= {(4^{2} - 2)}^{2} - 2 = 14^{2} - 2 = 194

Answered by malwan last updated on 24/Oct/20

x^{6} - 52 x^{3} + 1 = 0

t h i s e q u a t i o n h a s 6 s o l u t i o n s

4 c o m p l e x

a n d 2 r e a l x = 2 \pm \sqrt{3}

x = 2 + \sqrt{3}

\Rightarrow x^{4} + \frac{1}{x^{4}} = (97 + 56 \sqrt{3}) + (\frac{1}{97 + 56 \sqrt{3}})

= (97 + 56 \sqrt{3}) + (\frac{97 - 56 \sqrt{3}}{9409 - 9408})

= 97 + 97 = 194

a n d t h e s a m e w a y

x = 2 - \sqrt{3} \Rightarrow x^{4} + \frac{1}{x^{4}} = 194

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