Question

If a = 2 + √3, then find the value of (a-1/a)
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Answer 1

$\sf{\underline{\underline{\orange{Given}}}}$

• $\tt{a = 2 + \sqrt{3}}$

$\sf{\underline{\underline{\orange{To\: Find}}}}$

• $\tt{Value\: of\: a - \dfrac{1}{a}}$

$\sf{\underline{\underline{\orange{Solution}}}}$

★ It is given that :-

⟹ $\tt{a = 2 + \sqrt{3}}$

⟹ $\tt{\dfrac{1}{a} = \dfrac{1}{2 + \sqrt{3}}}$

★ Rationalising

⟹ $\tt{\dfrac{1}{a} = \dfrac{1}{2 + \sqrt{3}} × \dfrac{2 - \sqrt{3}}{2 - \sqrt{3}}}$

⠀⠀⟹ $\tt{\dfrac{1}{a} = \dfrac{2 - \sqrt{3}}{(2)^2 - (\sqrt{3})^2}}$

⠀⠀⠀⠀⟹ $\tt{\dfrac{1}{a} = \dfrac{2 - \sqrt{3}}{4 - 3}}$

⠀⠀⠀⠀⠀⠀⟹ $\tt{\dfrac{1}{a} = 2 - \sqrt{3}}$

★ Now, Putting the values

⟹ $\tt{a - \dfrac{1}{a} = (2 + \sqrt{3}) - (2 - \sqrt{3}}$

⠀⠀⟹ $\tt{a - \dfrac{1}{a} = 2 + \sqrt{3} - 2 + \sqrt{3}}$

⠀⠀⠀⠀⟹ $\tt{a - \dfrac{1}{a} = \sqrt{3} + \sqrt{3}}$

⠀⠀⠀⠀⠀⠀⟹ $\tt{a - \dfrac{1}{a} = 2\sqrt{3}}$