Question

If x = 2 - root 5 divided by 2 + root 5 and y = 2 + root 5 divided by 2- root 5 find the value of x plus y w

Answer 1

Answer:

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Answer 2

Answer:

${\bf Given :}\:\sf x =\dfrac{2-\sqrt{5}}{2+\sqrt{5}}\qquad y=\dfrac{2+\sqrt{5}}{2-\sqrt{5}}$

${\bf To\: Find :}\:\sf x+y$

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$\underline{\bigstar\:\boldsymbol{According\:to\:the\:Question :}}$

$:\implies\sf x+y\\\\\\:\implies\sf \dfrac{2-\sqrt{5}}{2+\sqrt{5}}+\dfrac{2+\sqrt{5}}{2-\sqrt{5}}\\\\{\scriptsize\qquad\bf{\dag}\:\:\texttt{Rationalization}}\\\\:\implies\sf \bigg\lgroup\dfrac{2-\sqrt{5}}{2+\sqrt{5}} \times \dfrac{2-\sqrt{5}}{2-\sqrt{5}}\bigg\rgroup+\bigg\lgroup\dfrac{2+\sqrt{5}}{2-\sqrt{5}} \times \dfrac{2+\sqrt{5}}{2+\sqrt{5}}\bigg\rgroup\\\\\\:\implies\sf \dfrac{(2 - \sqrt{5})^2}{(2)^2 -( \sqrt{5})^2} + \dfrac{(2 + \sqrt{5})^2}{(2)^2 -( \sqrt{5})^2}\\\\\\:\implies\sf \dfrac{(2 - \sqrt{5})^2 + (2+ \sqrt{5})^2}{(2)^2 -( \sqrt{5})^2}\\\\\\:\implies\sf \dfrac{4 + 5 -4 \sqrt{5} +4 + 5 +4 \sqrt{5}}{4 - 5}\\\\\\:\implies\sf \dfrac{18}{-1}\\\\\\:\implies\underline{\boxed{\sf - \:18}}$

⠀

$\therefore\:\underline{\textsf{Hence, Value of (x + y) will be \textbf{ - 18}}}.$