Question

if x=2 +√3 , Find the value of x^2 + 1/x^2​

Answer 1

Answer:

well first x²=[ 2+✓3]² = 7 + 4✓3

now 1/x² = 1/ 7+4✓3

using rationalisation

multiply and divide this term with 7-4✓3

hence 1/x²= 7-4✓3

therefore x²+1/x² = 7+4✓3 +7-4✓3= 14

Answer 2

Solution:-

:- Given

$\sf \implies \: x = 2 + \sqrt{ 3}$

:- To Find The value

$\sf \to \: {x}^{2} + \dfrac{1}{{x}^{2} }$

Now Take

$\sf \implies \: x = 2 + \sqrt{ 3}$

$\sf \implies \: \dfrac{1}{x} = \dfrac{1}{2 + \sqrt{3} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \implies \: \frac{1}{ \sf{x}} = \dfrac{1}{2 + \sqrt{3} } \times \dfrac{2 - \sqrt{3} }{2 - \sqrt{3} } \\ \\ \implies \sf \: \frac{1}{x} = \: \frac{2 - \sqrt{3} }{4 - 3} = \frac{2 - \sqrt{3} }{1}$

So We can write

$\sf \implies \: {x}^{2} + \dfrac{1}{{x}^{2} } = \bigg(x + \dfrac{1}{x} \bigg)^{2}$

Now put the value of x and 1/x

$\sf \implies \: (2 + \sqrt{3} + 2 - \sqrt{3} ) {}^{2}$

$\sf \implies \: (2 + \cancel{\sqrt{3}} + 2 - \cancel{\sqrt{3} }) {}^{2}$

$\rm \implies \: (4) {}^{2} = 16$

Answer is 16