Question

if x=√3 +1, find the value of x²+1/x²​

Answer 1

Solution:

Given,

• x = √3 + 1

To find ,

• x² + 1/x²

_________________

x = √3 + 1

By squaring on both sides

x² = (√3 + 1)²

Using

(a + b)² = a² + b² + 2ab

→ x² = (√3)² + 1² + 2(√3)(1)

→ x² = 3 + 1 + 2√3

→ x² = 4 + 2√3

→ x² = 2(2 + √3)

Substituting the value of x² in x² + 1/x²

→ [2(2 + √3)] + 1/[2(2 + √3)]

→ [4 + 2√3]² + 1/4 + 2√3

→ 4² + (2√3)² + 2(4)(2√3) + 1/4 + 2√3

→ 16 + 12 + 16√3 + 1/4 + 2√3

→ 29 + 16√3/4 + 2√3

By rationalizing the denominator

→ (29 + 16√3)(4 - 2√3)/(4 + 2√3)(4 - 2√3)

→ 116 - 58√3 + 64√3 - 96/4² - (2√3)²

→ 20 + 6√3/16 - 12

Taking common

→ 2(10 + 3√3)/4

→ 10 + 3√3/2

Hence, x² + 1/x² = 10 + 3√3/2

Answer 2

• Given:-

• x = √3+1

To Find:-

• The value of x² + 1/x²

Solution:-

x = √3+1

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

As we know that (a+b)² = a² + b² + 2ab

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

$\sf \implies {x}^{2} = {3} + {1} + 2 \sqrt{3}$

$\sf \implies {x}^{2} = 4+ 2 \sqrt{3}$

$\sf \implies {x}^{2} = 2(2 + \sqrt{3} )$

The value of x² + 1/x²

$\implies \sf {(4 + 2 \sqrt{3}) } + \dfrac{1}{4 + 2 \sqrt{3} }$

$\implies \sf \dfrac{ {(4 + 2 \sqrt{3} )}^{2} + 1}{4 + 2 \sqrt{3} }$

$\implies \sf \dfrac{ 16 + 12 + 16 \sqrt{3} + 1}{4 + 2 \sqrt{3} }$

$\implies \sf \dfrac{ 29 + 16 \sqrt{3} }{4 + 2 \sqrt{3} }$

Rationalising the denominator

$\implies \sf \dfrac{ 29 + 16 \sqrt{3} }{4 + 2 \sqrt{3} } \times \dfrac{4 - 2 \sqrt{3} }{4 - 2 \sqrt{3} }$

$\implies \sf \dfrac{ 116 - 58 \sqrt{3} + 64 \sqrt{3} - 96 }{16 - 12 }$

$\implies \sf \dfrac{ 20 + 6 \sqrt{3} }{4 }$

$\implies \sf \dfrac{ 2(10 + 3 \sqrt{3} ) }{4 }$

$\implies \sf \dfrac{ 10+ 3 \sqrt{3} }{2 }$

$\boxed{\sf {x}^{2} + \frac{1}{ {x}^{2} } = \dfrac{ 10 + 3 \sqrt{3} }{2 } }$