Question

$\frac{4+\sqrt{5}}{4-\sqrt{5}}+\frac{4-\sqrt{5}}{4+\sqrt{5}}=$ ___

• A. $2$
• B. $\frac{21}{11}$
• C. $\frac{42}{11}$
• D. $1$

Answer 1

The correct option is C

$\frac{42}{11}$

$\frac{4+\sqrt{5}}{4-\sqrt{5}}+\frac{4-\sqrt{5}}{4+\sqrt{5}}$

$=\frac{(4+\sqrt{5}{)}^2+(4-\sqrt{5}{)}^2}{(4-\sqrt{5})(4+\sqrt{5})}$$=\frac{2\left(\right(4{)}^2+\left(\sqrt{5}{)}^2\right)}{(4{)}^2-(\sqrt{5}{)}^2}\left[\begin{array}{c}(a+b{)}^2+(a-b{)}^2=2(a^2+b^2)\\ (a+b)(a-b)=a^2-b^2\end{array}\right]$

$=\frac{2(16+5)}{16-5}$

$=\frac{2\left(21\right)}{11}$

$=\frac{42}{11}$