Question

if x=5 + 2 root 6 and x = 1/p , then what will be the value of p2 + x2

Answer 1

$x = 5 + 2 \sqrt{6} \\ \frac{1}{x} = \frac{1}{5 + 2 \sqrt{6} } \times \frac{5 - 2 \sqrt{6} }{5 - 2 \sqrt{6} } \\ \frac{1}{x} = \frac{5 - 2 \sqrt{6} }{25 - 24} = \frac{5 - 2 \sqrt{6} }{1}$
Now
${p}^{2} + {x}^{2} = {(5 - 2 \sqrt{6)} }^{2} + {(5 + 2 \sqrt{6)} }^{2} \\ = {(5 - 2 \sqrt{6} + 5 + 2 \sqrt{6} )}^{2} - 2(5 + 2 \sqrt{6} )(2 - \sqrt{6} ) \\ = 100 - 2 = 98$