Question

if x=9-4√5,find x 2+1/X2​

Answer 1

☀️ Answer :  Value of x² + 1/x² is 322.

Given that : If x = 9 - 4√5

Need to find : Value of x² + 1/x²

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❍  Solution for the same is ::  

$\sf : \; \implies ( x )^{2} + \bigg( \dfrac{1}{x} \bigg)^{2}$

$\sf : \; \implies ( 9 - 4\sqrt{5} )^{2} + \bigg( \dfrac{1}{9-4\sqrt{5} } \bigg)^{2}$

$\sf : \; \implies ( 9 - 4\sqrt{5} )^{2} + \bigg( \dfrac{1}{9-4\sqrt{5}} \times \dfrac{9+ 4\sqrt{5}}{9 + 4 \sqrt{5}} \bigg)^{2}$

$\sf : \; \implies ( 9 - 4\sqrt{5} )^{2} + \bigg( \dfrac{1 \times [ \; 9 + 4 \sqrt{5} \; ]}{ [ \; 9-4\sqrt{5} \; ] \times [ \; 9 + 4 \sqrt{5} \; ] } \bigg)^{2}$

$\sf : \; \implies ( 9 - 4\sqrt{5} )^{2} + \bigg( \dfrac{9 + 4 \sqrt{5}}{ [ \; 9-4\sqrt{5} \; ] \times [ \; 9 + 4 \sqrt{5} \; ] } \bigg)^{2}$

$\sf : \; \implies ( 9 - 4\sqrt{5} )^{2} + \bigg( \dfrac{9 + 4 \sqrt{5}}{[ \; 9 \; ]^{2} - [ \; 4\sqrt{5} \; ]} \bigg)^{2}$

$\sf : \; \implies ( 9 - 4\sqrt{5} )^{2} + \bigg( \dfrac{9 + 4 \sqrt{5}}{[ \; 9 \; ]^{2} - [ 16 \times 5 ]} \bigg)^{2}$

$\sf : \; \implies ( 9 - 4\sqrt{5} )^{2} + \bigg( \dfrac{9 + 4 \sqrt{5}}{81- 80} \bigg)^{2}$

$\sf : \; \implies ( 9 - 4\sqrt{5} )^{2} + \bigg( \dfrac{9 + 4 \sqrt{5}}{1} \bigg)^{2}$

$\sf : \; \implies ( 9 - 4\sqrt{5} )^{2} + ( 9 + 4 \sqrt{5} )^{2}$

$\sf : \; \implies \{ 9^{2} - 72\sqrt{5} + 16(\sqrt{5})^{2} \} + ( 9 + 4 \sqrt{5} )^{2}$

$\sf : \; \implies \{ 81 - 72\sqrt{5} + 16 \times 5 \} + ( 9 + 4 \sqrt{5} )^{2}$

$\sf : \; \implies \{ 81 - 72\sqrt{5} + 80 \} + ( 9 + 4 \sqrt{5} )^{2}$

$\sf : \; \implies \{ 161 - 72\sqrt{5} \} + ( 9 + 4 \sqrt{5} )^{2}$

$\sf : \; \implies \{ 161 - 72\sqrt{5} \} + \{ 9^{2} + 72\sqrt{5} + 16( \sqrt{5})^{2} \}$

$\sf : \; \implies \{ 161 - 72\sqrt{5} \} + \{ 81 + 72\sqrt{5} + 16 \times 5 \}$

$\sf : \; \implies \{ 161 - 72\sqrt{5} \} + \{ 81 + 72\sqrt{5} + 80 \}$

$\sf : \; \implies \{ 161 - 72\sqrt{5} \} + \{161 + 72\sqrt{5} \}$

$\sf : \; \implies 161 - 72\sqrt{5} + 161 + 72\sqrt{5}$

$\sf : \; \implies ( 161 + 161 ) + ( 72\sqrt{5}- 72\sqrt{5} )$

$\sf : \; \implies 322 + 0$

$\boxed{\bf{ \bigstar \;\; x^{2} + \dfrac{1}{x^{2}} = 322 }}$

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• Henceforth, the value of  x² + 1/x² is 322.