Determine-x-4-1-x-4-if-x-2-1-x-2-34-

Question Number 8148 by Rasheed Soomro last updated on 02/Oct/16

Determine x^{4} - \frac{1}{x^{4}}, if x^{2} + \frac{1}{x^{2}} = 34 .

Commented by sou1618 last updated on 02/Oct/16

s e t X = x^{2} ⩾ 0

X + \frac{1}{X} = 34

{(X - \frac{1}{X})}^{2} = {(X + \frac{1}{X})}^{2} - 4 = 34^{2} - 2^{2} = 36 \times 32

\Rightarrow

X - \frac{1}{X} = \pm 24 \sqrt{2}

X^{2} - \frac{1}{X^{2}} = (X + \frac{1}{X}) (X - \frac{1}{X}) = 34 \times (\pm 24 \sqrt{2})

= \pm 816 \sqrt{2}

Commented by Rasheed Soomro last updated on 02/Oct/16

Thank S! G \underset{\overset{}{⌣}}{\overset{⌢}{∙}} \overset{⌢}{∙} D approach!

Answered by trapti rathaur@ gmail.com last updated on 02/Oct/16

Missing \left or extra \right

34 \times 34 = x^{4} + \frac{1}{x^{4}} + 2

x^{4} + \frac{1}{x^{4}} = 1154

Missing \left or extra \right

Missing \left or extra \right

{(1154)}^{2} = {(x^{4} - \frac{1}{x^{4}})}^{2} + 4

{(x^{4} - \frac{1}{x^{4}})}^{2} = {(1154)}^{2} - {(2)}^{2}

(x^{4} - \frac{1}{x^{4}}) = \sqrt{1331712}

= 816 \sqrt{2}

A N S W E R -

(x^{4} - \frac{1}{x^{4}}) = 816 \sqrt{2}

Commented by Rasheed Soomro last updated on 02/Oct/16

Thank S!

You lost one answer by taking only + ve

squareroot of 1331712 : (x^{4} - \frac{1}{x^{4}}) = \sqrt{1331712}

You should have written

(x^{4} - \frac{1}{x^{4}}) = \pm \sqrt{1331712} = \pm 816 \sqrt{2}

Commented by trapti rathaur@ gmail.com last updated on 02/Oct/16

t h a n k s . i f o r g e t i t

Answered by Rasheed Soomro last updated on 02/Oct/16

x^{2} + \frac{1}{x^{2}} = 34, x^{4} - \frac{1}{x^{4}} = ?

Approach :

x^{2} + \frac{1}{x^{2}} \to {\begin{cases} x + \frac{1}{x} \\ x - \frac{1}{x} \end{cases} \to x^{2} - \frac{1}{x^{2}} \land x^{2} + \frac{1}{x^{2}} \to x^{4} - \frac{1}{x^{4}}

x^{2} + \frac{1}{x^{2}} = 34

x^{2} + 2 + \frac{1}{x^{2}} = 34 + 2 \land x^{2} - 2 + \frac{1}{x^{2}} = 34 - 2

{(x + \frac{1}{x})}^{2} = {(\pm 6)}^{2} \land {(x - \frac{1}{x})}^{2} = {(\pm 4 \sqrt{2})}^{2}

x + \frac{1}{x} = \pm 6 \land x - \frac{1}{x} = \pm 4 \sqrt{2}

\Rightarrow x^{2} - \frac{1}{x^{2}} = \pm 24 \sqrt{2}

\begin{matrix} x^{2} - \frac{1}{x^{2}} = \pm 24 \sqrt{2} \\ \times \\ x^{2} + \frac{1}{x^{2}} = 34 (given) \end{matrix}} \Rightarrow x^{4} - \frac{1}{x^{4}} = \pm 816 \sqrt{2}