Question

If $x=\frac{\sqrt{5}-2}{\sqrt{5}+2}$ and $y=\frac{\sqrt{5}+2}{\sqrt{5}-2}$, find the value of $x^2+y^2-xy$.

Answer 1

$x=\frac{\sqrt{5}-2}{\sqrt{5}+2}\times \frac{\sqrt{5}-2}{\sqrt{5}-2}=\frac{5+4-4\sqrt{5}}{5-4}=9-4\sqrt{5}$

$y=\frac{\sqrt{5}+2}{\sqrt{5}-2}\times \frac{\sqrt{5}+2}{\sqrt{5}+2}=\frac{5+4+4\sqrt{5}}{5-4}=9+4\sqrt{5}$

$x^2+y^2-xy$

$={(x+y)}^2-2xy-xy$

$={(x+y)}^2-3xy$

$={(9-4\sqrt{5}+9+4\sqrt{5})}^2-3(9-4\sqrt{5})(9+4\sqrt{5})$

$=324-3(81-16\times 5)$

$=324-3(81-80)$

$=324-3\times 1$

$=324-3=321$