Question

If $x={(2+\sqrt{5})}^{1/2}+{(\sqrt{5}-2)}^{1/2}$ and $y={(2+\sqrt{5})}^{1/2}-{(\sqrt{5}-2)}^{1/2}$,then evaluate $x^2+y^2$.

Answer 1

$a$ $b$

$\uparrow$ $\uparrow$

$x=(2+\sqrt{5}{)}^{1/2}+(\sqrt{5}-2{)}^{1/2}$

$y=\left(\right(2+\sqrt{5}){)}^{1/2}-(\sqrt{5}-2{)}^{1/2}$

i.e $x=a^{1/2}+b^{1/2}$

$y=a^{1/2}-b^{1/2}$

So $x^2+y^2=a+b+2a^{1/2}{b}^{1/2}$

$+a+b-2a^{1/2}{b}^{1/2}$

$=2(a+b)$

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

$=2.2\sqrt{5}$

$=4\sqrt{5}$