Question

if x = 2 + √3 , find the value of ( x² + 1 / x² )^2


( don't give unnecessary answers ) ( give correct answer only )​

Answer 1

x = 2 + √3

x² = ( 2 + √3)²

= 2²+ 4√3 + 3

= 4 + 4√3 + 3

= 7 + 4√3

(x² + 1/x²)² = (7 +4√3 +( 1 / (7+4√3)))²

= (7+4√3)² + (2 .7+4√3.1/7+4√3) +( 1/7+4√3)²

= 7²+(4√3)²+56√3 + 2 + [1/(7²+8√3 +( 4√3)²]

= 49 + 48 + 56√3 + 2 + [1/( 97 + 56√3)]

= 97+56√3 +( 1 / (97 +56✓3) )+ 2

on substituting the values of √3, we get the answer 196

Answer 2

$\large\underline{\sf{Given- }}$

$\red{\rm :\longmapsto\:x = 2 + \sqrt{3}}$

$\large\underline{\sf{To\:Find - }}$

$\red{\rm :\longmapsto\: {\bigg( {x}^{2} + \dfrac{1}{ {x}^{2} } \bigg) }^{2}}$

Solution :-

Given that

${\rm :\longmapsto\:x = 2 + \sqrt{3}}$

So,

$\rm :\longmapsto\:\dfrac{1}{x}$

$\rm \: = \: \: \dfrac{1}{2 + \sqrt{3} }$

☆ On rationalizing the denominator, we get

$\rm \: = \: \: \dfrac{1}{2 + \sqrt{3} } \times \dfrac{2 - \sqrt{3} }{2 - \sqrt{3} }$

$\rm \: = \: \: \dfrac{2 - \sqrt{3} }{ {(2)}^{2} - {( \sqrt{3})}^{2} }$

$\rm \: = \: \: \dfrac{2 - \sqrt{3} }{4 - 3}$

$\rm \: = \: \: \dfrac{2 - \sqrt{3} }{1}$

$\rm \: = \: \: 2 - \sqrt{3}$

$\bf\implies \:\dfrac{1}{x} = 2 - \sqrt{3}$

Consider,

$\bf :\longmapsto\: {x}^{2} + \dfrac{1}{ {x}^{2} }$

$\rm \: = \: \: {\bigg(x + \dfrac{1}{x} \bigg) }^{2} - 2 \times x \times \dfrac{1}{x}$

$\red{\bigg \{ \because \: {x}^{2} + {y}^{2} = {(x + y)}^{2} - 2xy\bigg \}}$

$\rm \: = \: \: {\bigg(x + \dfrac{1}{x} \bigg) }^{2} - 2$

[☆ On substituting the values, we get]

$\rm \: = \: \: {\bigg(2 + \sqrt{3} + 2 - \sqrt{3} \bigg) }^{2} - 2$

$\rm \: = \: \: {\bigg(2 + 2\bigg) }^{2} - 2$

$\rm \: = \: \: {\bigg(4\bigg) }^{2} - 2$

$\rm \: = \: \: 16 - 2$

$\rm \: = \: \: 14$

$\purple{ \boxed{\bf :\implies\: {x}^{2} + \dfrac{1}{ {x}^{2}} = 14 }}$

Now,

Consider,

${\bf :\longmapsto\: {\bigg( {x}^{2} + \dfrac{1}{ {x}^{2} } \bigg) }^{2}}$

$\rm \: = \: \: {(14)}^{2}$

$\rm \: = \: \: 196$

Hence,

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \underbrace{ \boxed {\bf \: {\bigg( {x}^{2} + \dfrac{1}{ {x}^{2} } \bigg) }^{2} = 196}}$

More Identities to know:

• (a + b)² = a² + 2ab + b²

• (a - b)² = a² - 2ab + b²

• a² - b² = (a + b)(a - b)

• (a + b)² = (a - b)² + 4ab

• (a - b)² = (a + b)² - 4ab

• (a + b)² + (a - b)² = 2(a² + b²)

• (a + b)³ = a³ + b³ + 3ab(a + b)

• (a - b)³ = a³ - b³ - 3ab(a - b)