Math, asked by hajiabdullahcongress, 7 months ago

if x-1/X=3 find the value of x^2+1/x^2 and x=^4+1/x^4​

Answers

Answered by Darkrai14
4

Correct Question:-

If \rm x-\dfrac{1}{x}=3 , find the value of \rm x^2+\dfrac{1}{x^2} \ and \ x^4 +\dfrac{1}{x^4}

Solution:-

\rm x- \dfrac{1}{x}=3

Squaring it will give,

\rm\implies \Bigg ( x- \dfrac{1}{x} \Bigg )^2 =(x)^2+\Bigg ( \dfrac{1}{x}\Bigg )^2-2 \times x \times \dfrac{1}{x} \qquad\qquad ...[ since, \ (a-b)^2=a^2+b^2-2ab]

Solving it,

\rm\implies (3)^2 =x^2+\dfrac{1}{x^2} -2

\rm \implies 9 =x^2+ \dfrac{1}{x^2}-2

\rm \implies 9+2 =x^2+ \dfrac{1}{x^2}

\rm \implies 11 =x^2+ \dfrac{1}{x^2}

\qquad\qquad\qquad\bigstar\boxed{\bf x^2+ \dfrac{1}{x^2}=11}\bigstar

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Now we will find \rm x^4 + \dfrac{1}{x^4}

We know that,

\boxed{\rm (a^2+b^2)^2 = a^4+b^4+ 2a^2b^2}

Using this identity,

\rm\implies\Bigg ( x^2+\dfrac{1}{x^2} \Bigg )^2 = x^4 + \dfrac{1}{x^4} + 2 \times x^2 \times \dfrac{1}{x^2}

\rm\implies (11 )^2 = x^4 + \dfrac{1}{x^4} + 2

\rm\implies 121 = x^4 + \dfrac{1}{x^4} + 2

\rm\implies 121-2= x^4 + \dfrac{1}{x^4}

\rm\implies 119= x^4 + \dfrac{1}{x^4}

\qquad\qquad\qquad\bigstar\boxed{\bf x^4+ \dfrac{1}{x^4}=119}\bigstar

Answered by rajeevr06
2

Answer:

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

now,

 {x}^{2}  +  \frac{1}{ {x}^{2} }  = (x -  \frac{1}{x}) {}^{2}   + 2 \times x \times  \frac{1}{x}  =  {3}^{2}  + 2 = 9 + 2 = 11 \:  \: ans.

now,

 {x}^{4}  +  \frac{1}{ {x}^{4} }  = ( {x}^{2}  +  \frac{1}{ {x}^{2} } ) {}^{2}  - 2 {x}^{2}  \times  \frac{1}{ {x}^{2} }  =  {11}^{2}  - 2 = 121 - 2 = 119

Thanks.

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