Question

if X equal to root 3 minus root 2 upon root 3 + root 2 and Y equal to root 3 + root 2 upon root 3 minus root 2 then find the value of x square + y square + x y

Answer 1

Answer:99

Step-by-step explanation:

as given

$x= \frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}\\\\x^{2}=(\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}})^{2}\\\\=\frac{3+2+2\sqrt{6}}{3+2-2\sqrt{6}}\\\\=\frac{5+2\sqrt{6}}{5-2\sqrt{6}}$

and

$y= \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}\\\\y^{2}=(\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}})^{2}\\\\=\frac{3+2-2\sqrt{6}}{3+2+2\sqrt{6}}\\\\=\frac{5-2\sqrt{6}}{5+2\sqrt{6}}$

and xy =1

so the value of  

$x^{2}+y^{2}+xy\\\\=\frac{5+2\sqrt{6}}{5-2\sqrt{6}}+\frac{5-2\sqrt{6}}{5+2\sqrt{6}}+1\\\\=\frac{(5+2\sqrt{6})^{2}+(5-2\sqrt{6})^{2}}{(5-2\sqrt{6})*(5+2\sqrt{6})}+1\\\\=\frac{2(25+24)}{25-24}+1\\\\=98+1\\\\=99$