Question

If a=2+√3, then a+1/a=?


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Answer 1

GIVEn

If a=2+√3, then a+1/a=?

SOLUTIOn

• If a = 2 + √3

Then,

→ 1/a = 1/2 + √3

★ Rationalise its denominator

→ 1/a = 1/2 + √3 × 2 - √3/2 - √3

→ 1/a = 2 - √3/(2)² - (√3)²

→ 1/a = 2 - √3/4 - 3

→ 1/a = 2 - √3

So, the value of a + 1/a

• a = 2 + √3
• 1/a = 2 - √3

→ a + 1/a

→ 2 + √3 + 2 - √3

→ 4

Answer 2

$\huge\blue{\tt{Question:-}}$

$If \: a = (2 + \sqrt{3} ) , \: then \: a + \frac{1}{a} = ?$

$\huge\blue{\tt{Solution:-}}$

$a + \frac{1}{a} = (2 + \sqrt{3} ) + \frac{1}{(2 + \sqrt{3} )}$

$solving \: \frac{1}{a}$

Using rationalisation,

$\frac{1}{(2 + \sqrt{3} )} \times \frac{(2 - \sqrt{3} )}{(2 - \sqrt{3} )}$

$Since, (a+b)(a-b) = {a}^{2} - {b}^{2}$

Therefore,

$= > \frac{(2 - \sqrt{3}) }{ {(2)}^{2} - {( \sqrt{3} )}^{2} }$

$= > \frac{(2 - \sqrt{3}) }{4 - 3}$

$= > \frac{(2 - \sqrt{3}) }{1} = (2 - \sqrt{3} )$

$Now, a + \frac{1}{a} = (2 + \sqrt{3} ) + (2 - \sqrt{3} )$

=> 2 + 2

=> 4.

Hence, 4 is the required answer.