Question

If x= 2+ ✓3
then
find
the value of x² + 1 /x²
​

Answer 1

Given:

• x = 2 + √3

To find:

$\bullet \quad \sf{x^{2} + \dfrac{1}{x^{2} }}$

Method:

$\sf{\dfrac{1}{x} = \dfrac{1}{2+\sqrt{3}} \times \dfrac{2 - \sqrt{3} }{2 - \sqrt{3} } = 2 - \sqrt{3} }$

Now let us find x + 1/x

⇒ x + 1/x = 2 + √3 + 2 - √3

⇒ x + 1/x = 4

Squaring on both sides,

(x + 1/x)² = 4²

$\implies \sf{x^{2} + \dfrac{1}{x^{2}} + 2(x)(\dfrac{1}{x}) = 16}\\\\\\\implies \sf{x^{2} + \dfrac{1}{x^{2}}+ 2= 16}\\\\\\\implies \sf{x^{2} + \dfrac{1}{x^{2}} = 16 - 2}\\\\\\\implies \boxed{\bf{x^{2} + \dfrac{1}{x^{2} } = 14 }}$

The algebraic identity used here is

(a + b)² = a² + b² + 2ab