If-x-3-x-2-x-1-x-3-1-x-2-1-x-70-x-4-1-x-4-

Question Number 172326 by nurtani last updated on 25/Jun/22

I f

(x^{3} + x^{2} + x) + (\frac{1}{x^{3}} + \frac{1}{x^{2}} + \frac{1}{x}) = 70

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

Commented by infinityaction last updated on 25/Jun/22

x^{3} + \frac{1}{x^{3}} + x^{2} + \frac{1}{x^{2}} + x + \frac{1}{x} = 70

{(x + \frac{1}{x})}^{3} - 3 (x + \frac{1}{x}) + {(x + \frac{1}{x})}^{2} - 2 + x + \frac{1}{x} = 70

l e t x + \frac{1}{x} = a

a^{3} + a^{2} - 2 a - 72 = 0

(a - 4) (a^{2} + 5 a + 18) = 0

a = 4, - \frac{5}{2} \pm \frac{\sqrt{47} i}{2}

a = 4

s o

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

x^{2} + \frac{1}{x^{2}} = 14

x^{4} + \frac{1}{x^{4}} = 194

Commented by nurtani last updated on 25/Jun/22

T h a n k y o u s i r!

Commented by Tawa11 last updated on 25/Jun/22

Great sir

Commented by peter frank last updated on 27/Jun/22

thank you

Answered by greougoury555 last updated on 25/Jun/22

(x^{2} + x + 1) (\frac{1}{x^{2}} + \frac{1}{x} + 1) = 70

(x^{2} + x + 1) (\frac{x^{2} + x + 1}{x^{2}}) = 70

{(x^{2} + x + 1)}^{2} = 70 x^{2}

(x^{2} + x + 1 - x \sqrt{70}) (x^{2} + x + x \sqrt{70} + 1) = 0

{\begin{cases} x^{2} + (1 - \sqrt{70}) x + 1 = 0 \\ x^{2} + (1 + \sqrt{70}) x + 1 = 0 \end{cases}

Answered by Rasheed.Sindhi last updated on 26/Jun/22

x^{2} (x + 1 + \frac{1}{x}) + \frac{1}{x^{2}} (\frac{1}{x} + 1 + x) = 70

(x + 1 + \frac{1}{x}) (x^{2} + \frac{1}{x^{2}}) = 70

(x + \frac{1}{x} + 1) ({(x + \frac{1}{x})}^{2} - 2) = 70

(y + 1) (y^{2} - 2) = 70; x + \frac{1}{x} = y

y^{3} + y^{2} - 2 y - 72 =

(y - 4) (y^{2} + 5 y + 18) = 0

y = 4 ∣ y = \frac{- 5 \pm \sqrt{25 - 72}}{2} = \frac{- 5 \pm i \sqrt{47}}{2}

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

x^{2} + \frac{1}{x^{2}} = 16 - 2 = 14

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