Question

x^2+1/x^2=14 , thenfind the value of x^2-1/x^2

Answer 1

( x + 1/x )² = x² + 1/x² + 2 × x × 1/x

( x + 1/x )² = 14+2 = 16

x + 1/x = √16 = 4 -------------1

now ,
( x - 1/x )² = x² + 1/x² - 2 × x × 1/x

( x - 1/x )² = 14-2

x - 1/x = √12 = 2√3 -------------2

adding equation 1 and 2 ,


x + 1/x + x - 1/x = 4 + 2√3

2x = 4 + 2√3

x = 2 + √3

therefore

x^2-1/x^2 = (2 + √3)² - 1/(2 + √3)²

x² - 1/x² =[ (7 + 4√3 ) (7 + 4√3 ) -1 ]/(7 + 4√3 )

x² - 1/x² = (97 + 56√3 -1 ) / (7 + 4√3 )

x² - 1/x² =( 96 + 56√3 ) / (7 + 4√3 )