Question

$\text{If}y=\frac{\sqrt{5}+2}{\sqrt{5}-2}$, then $y^2=$ _________.

• A. $161+72\sqrt{5}$
• B. $161-72\sqrt{5}$
• C. $161$
• D. $144\sqrt{5}$

Answer 1

The correct option is A

$161+72\sqrt{5}$

$y=\frac{\sqrt{5}+2}{\sqrt{5}-2}$

$y=\frac{\sqrt{5}+2}{\sqrt{5}-2}\times \frac{\sqrt{5}+2}{\sqrt{5}+2}$

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

$y=\frac{(\sqrt{5}{)}^2+2(\sqrt{5}\left)\right(2)+(2{)}^2}{5-4}$

$=\frac{5+4\sqrt{5}+4}{1}$

$=9+4\sqrt{5}$

$y^2=(9+4\sqrt{5}{)}^2$

$=(9{)}^2+2(9\left)\right(4\sqrt{5})+(4\sqrt{5}{)}^2$

$=81+72\sqrt{5}+80$

$=161+72\sqrt{5}$