Question

if x =9-4root5,findthe value of x square - 1/xsquare

Answer 1

$x = 9 - 4 \sqrt{5} \\ \frac{1}{x} = \frac{1}{ 9 - 4 \sqrt{5} } = \frac{{ 9 + 4 \sqrt{5} } }{ (9 - 4 \sqrt{5})( { 9 + 4 \sqrt{5} )} } \\ = \frac{9 + 4 \sqrt{5} }{81 - 80} = 9 + 4 \sqrt{5}$

$x + \frac{1}{x} = (9 - 4 \sqrt{5} ) + (9 + 4 \sqrt{5} ) \\ = 18 \\ x - \frac{1}{x} = (9 - 4 \sqrt{5} ) - (9 + 4 \sqrt{5} ) \\ = - 8 \sqrt{5}$

${x}^{2} - \frac{1}{ {x}^{2} } = (x + \frac{1}{x} )(x - \frac{1}{x} ) \\ = 18 \times - 8 \sqrt{5} = - 144 \sqrt{5}$