Question

If x=9+4root 5,find the value of x sq.+1by x sq.

Answer 1

$x = 9 + 4 \sqrt{5} \\ \frac{1}{x} = \frac{1}{9 + 4 \sqrt{5} } \\ \frac{1}{x} = \frac{9 - 4 \sqrt{5} }{(9 + 4 \sqrt{5} )(9 - 4 \sqrt{5} )} = \frac{9 - 4 \sqrt{5} }{81 - 80} = 9 - 4 \sqrt{5} \\ {x}^{2} + \frac{1}{ {x}^{2} } = {(x + \frac{1}{x} )}^{2} - 2 \\ ={ (9 + 4 \sqrt{5} + 9 - 4 \sqrt{5} ) }^{2} - 2 \\ = {18}^{2} - 2 = 324 - 2 = 322$
a square +b square =(a+b)square - 2ab
because (a+b)square - 2ab = a square + b square+2ab-2ab= a square + b square.