Question

if x=5-2root6 find x^2+1/x^2

Answer 1

Here

$x = 5 - 2\sqrt 6$

$\frac{1}{x^2} = \frac {1}{5-2\sqrt 6}$

Rationalising the denominator by multiplying Numerator and Denominator by ( 5 + 2√6 )

We need to find

$x^2 + \frac{1}{x^2}$

$\frac{1}{5-2\sqrt6} \times \frac{5+2\sqrt6}{5+2\sqrt 6} \\=\frac{5+2\sqrt6}{ {5}^2 - (2\sqrt 6)^2}$

$\frac{1}{x}=\frac{5+2\sqrt6}{25-24}= 5+2\sqrt6$

Now, we know that

$(x + \frac{1}{x})^2 =x^2 +\frac{1}{x^2}+2(x)(\frac{1}{x})$

Putting values of x and 1/x in LHS

$[(5-2\sqrt6)+(5+2\sqrt6)]^2 =x^2 +\frac{1}{x^2}+2 \\(10)^2 -2=x^2 +\frac{1}{x^2} \\x^2+\frac{1}{x^2} = 100 -2$

$x^2+\frac{1}{x^2} = 98$