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If x=2+root5, prove thatx^2+1/x^2=18

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If x=2+root5, prove thatx^2+1/x^2=18

    • Posted by Vikash Kumar 4 years, 5 months ago

    CBSE > Class 09 > Mathematics
    • 1 answers

    Rajan Kumar Pasi 4 years, 5 months ago

    {tex}Q.\text{ If x = 2 + }\sqrt5\text{, prove that: }{x^2+\frac1{x^2}=18.}{/tex}
     

    {tex}Sol.\text{ If x = 2 + }\sqrt5,\\ {=>\ x^2=4+4\sqrt5+5= 9+4\sqrt5}\\ So,{x^2+\frac1{x^2}={(9+4\sqrt5)+\frac1{(9+4\sqrt5)}}}\\ (Rationalising,)\\ =>\ {(9+4\sqrt5)+\frac1{(9+4\sqrt5)}\times\frac{(9-4\sqrt5)}{(9-4\sqrt5)}}\\ =>\ {(9+4\sqrt5)+\frac{(9-4\sqrt5)}{(81-80)}}\\ =>\ {(9+4\sqrt5)+\frac{(9-4\sqrt5)}{(1)}}\\ =>\ {(9+4\sqrt5)+{(9-4\sqrt5)}}\\ =>18\\ =>LHS=RHS\\ Hence Proved...{/tex} {tex}\huge{Identities\ used:}\\ \huge{1.\ (a+b)^2=a^2+2ab+b^2}\\ \huge{2.\ (a+b)(a-b)=a^2-b^2}\\{/tex}

     

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