Question

If x + 1/x = 2, find x⁴ + 1/x⁴​

Answer 1

$\Large{\underline{\bf{ Given}} : - }$

$\bullet \: \tt x + \dfrac{1}{x} = 2$

$\Large{\underline{\bf{ To \: find }} : - }$

$\bullet \: \tt value \: of \: x^4 + \dfrac{1}{x^4 }$

$\Large{\underline{ \bf{Solution}} : -}$

$\sf : \implies x + \dfrac{1}{x} = 2$

$\sf :\implies x + \dfrac{1}{x } - 2 = 0$

$\rm By \: taking \: LCM$

$\sf :\implies\dfrac{ x^2 + 1 - 2x }{x } = 0$

$\textrm{ By cross multiplication on both sides }$

$\sf :\implies x^2 + 1 - 2x = 0 \times x$

$\sf :\implies \sf x^2 - 2x + 1 = 0$

$\sf By\: comparing \: the \: given \: quadratic \: equation \: with \: ax^2 + bx + c = 0$

We have,

• a = 1
• b = - 2
• c = 1

$\tt Using \: the \: Quadratic \: formula :$

$\tt{ Formula : -} \underline{\boxed{\tt{ x = \dfrac{ - b \pm \sqrt{ b^2 - 4ac }}{ 2a }}}}$

$\sf{: \implies x = \dfrac{ - ( - 2 ) \pm \sqrt{ ( - 2 )^2 - 4 ( 1 )( 1 )}}{ 2 ( 1 )}}$

$\sf :\implies x = \dfrac{ 2 \pm \sqrt{4 - 4 ( 1 ) } }{ 2 }$

$\sf :\implies x = \dfrac{ 2 \pm \sqrt{4 - 4 } }{2 }$

$\sf :\implies x = \dfrac{2 \pm \sqrt{0} }{2}$

$\sf :\implies x = \dfrac{2 \pm 0 }{2}$

$\sf :\implies x = \cancel{\dfrac{2 }{2}}$

$\sf :\implies x = 1$

$\tt Now, \: let's \: find \: out \: the \: value \: of \: x^4 +\dfrac{ 1}{ x^4 }$

• $\sf x = 1$

$\sf :\implies { x}^4 = ( 1 )^4 = 1$

$\sf Now, for \: value \: of \: {x}^4 + \dfrac{1}{ {x}^4} :$

$\sf :\implies 1 + \dfrac{1}{1}$

$\sf :\implies 1 + 1$

$\sf :\implies 2$

$\sf \\ \\ \bf \pink{ \therefore Value \: of \: x^4 + \dfrac{1}{x^4 } = 2 }$