Question

if x = √3-√2 then find the value of ( x-1/x )^2 .................tell me the answer pls ​

Answer 1

Answer:

$\underline{\underline{\large{\sf Given\:that:}}}$

x = √3 - √2

$\sf \implies \frac{1}{x} = \frac{1}{ \sqrt{3} - \sqrt{2} } \\ \\ \sf \implies \frac{1}{x} = \frac{1}{ \sqrt{3} - \sqrt{2} } \times \frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} + \sqrt{2} } \\ \\ \sf \implies \frac{1}{x} = \frac{ \sqrt{3} + \sqrt{2} }{( \sqrt{3}) {}^{2} - ( \sqrt{2}) {}^{2} } \\ \\ \sf \implies \frac{1}{x} = \frac{ \sqrt{3} + \sqrt{2} }{3 - 2} \\ \\ \sf \implies \frac{1}{x} = \sqrt{3} + \sqrt{2}$

Now,

$\sf x - \frac{1}{x} = \sqrt{3} - \sqrt{2} - ( \sqrt{3} + \sqrt{2} ) \\ \\ \sf \implies \sqrt{3} - \sqrt{2} - \sqrt{3} - \sqrt{2} \\ \\ \sf \implies - 2 \sqrt{2}$

We have got-

$\sf x - \frac{1}{x} = - 2 \sqrt{2} \\ \\ \underline{\underline{ \sf Squaring \: both \: sides - }} \\ \\ \sf (x - \frac{1}{x} ) {}^{2} = ( - 2 \sqrt{2} ) {}^{2} \\ \\ \boxed{ \red{\bf (x - \frac{1}{x} ) {}^{2} =8}}$

Hence, the answer is 8.

Answer 2

Lets first, do operation on the equation so that it will be easy to do the sum.

We will use the identity

$</p><p>(a-b)^2=a^2+b^2-2ab\\</p><p>$

Now we'll put the value of x in the equation.

After that, we will rationalise the denominator by multiplying the numerator and denominator by

$</p><p>5+2\sqrt{6}\\</p><p>$

You'll get the answer as

$</p><p>6+2\sqrt{6}-2\sqrt{3}-2\sqrt{2}</p><p>$