यदि `x = 2 + sqrt(3)` तो `x^(2)+(1)/(x^(2))` का मान ज्ञात करें।

Explanation

प्रश्न से, `x = 2 + sqrt(3)`
`therefore (1)/(x)=(1)/(2+sqrt(3))=(1)/(2+sqrt(3))xx(2-sqrt(3))/(2-sqrt(3))`
`=(2-sqrt(3))/(2^(2)-(sqrt3)^(2))=(2-sqrt(3))/(4-3)=2-sqrt(3)`
अब, `x^(2)+(1)/(x^(2))=(2+sqrt(3))^(2)+(2-sqrt(3))^(2)`
`=2^(2)+(sqrt(3))^(2)+2xx2sqrt(3)+2^(2)+(sqrt(3))^(2)-2xx2sqrt(3)`
`=4+3+4sqrt(3)+4+3-4sqrt(3)=14`
दूसरी विधि : प्रश्न से, `x = 2 + sqrt(3)`
`therefore (1)/(x)=(1)/(2+sqrt(3))=(1)/(2+sqrt(3))xx(2-sqrt(3))/(2-sqrt(3))=(2-sqrt(3))/(2^(2)-(sqrt(3))^(2))=(2-sqrt(3))/(4-3)=2-sqrt(3)`
अब, `x^(2)+(1)/(x^(2))=(x+(1)/(x))^(2)-2x.(1)/(x)=(2+sqrt(3)+2-sqrt(3))^(2)-2`
`=4^(2)-2=16-2=14`