Question

if a=2+√6 and b=1/a how we will find the value of a2-b2?

Answer 1

$a = 2 + \sqrt{6} \\ \\ \\ \\ b = \frac{1}{2 + \sqrt{6} } (rationalise) \\ \\ \\ \\ \implies b = \frac{1(2 - \sqrt{6}) }{2 + \sqrt{6}(2 - \sqrt{6} ) } \\ \\ \\ \\ \implies b = \frac{2 - \sqrt{6} }{4 - 6} \\ \\ \\ \implies \boxed{b = \frac{2 - \sqrt{6} }{ - 2} }$

Now we got the value of a and b so we can simplify a^2 - b^2

${a}^{2} - {b}^{2} \\ \\ \\ = {(2 + \sqrt{6} )}^{2} - { (\frac{2 - \sqrt{6} }{ - 2} )}^{2} \\ \\ \\ = 4 + 6 + 4 \sqrt{6} - ( \frac{4 + 6 - 4 \sqrt{6} }{ - 2} ) \\ \\ \\ = 10 + 4 \sqrt{6} - ( \frac{10 - 4 \sqrt{6} }{ - 2} ) \\ \\ \\ = 10 + 4 \sqrt{6} - ( \frac{4 \sqrt{6} - 10 }{2} ) \\ \\ \\ \\ = \frac{20 + 8 \sqrt{6} - 4 \sqrt{6 } + 10}{2} \\ \\ \\ \\ = \frac{30 + 4 \sqrt{6} }{2} \\ \\ \\ = \boxed{15 + 2 \sqrt{6} }$

We got 15 + 2 root 6 as our simplified answer.