Question

X=√3+1/√3-1 y=√3-1/√3+1
x²+y²+xy

Answer 1

$x = \frac{\sqrt{3} + 1}{\sqrt{3} - 1} , \:and \\y= \frac{\sqrt{3} - 1}{\sqrt{3} + 1}$

$x + y = \frac{\sqrt{3} + 1}{\sqrt{3} - 1} + \frac{\sqrt{3} - 1}{\sqrt{3} + 1}$

$= \frac{ (\sqrt{3}+1)^{2}+(\sqrt{3}-1)^{2}}{(\sqrt{3}-1)(\sqrt{3}+1)}$

$= \frac{2[(\sqrt{3})^{2} + 1^{2} )}{(\sqrt{3})^{2} - 1^{2})}\\= \frac{2(3+1)}{(3-1)}\\= \frac{2 \times 4}{2} \\= 4 \:---(1)$

$xy = \big(\frac{\sqrt{3} + 1}{\sqrt{3} - 1}\big) \big(\frac{\sqrt{3} - 1}{\sqrt{3} + 1}\big) \\= 1 \:--(2)$

$Now , \red { x^{2} + y^{2} + xy} \\= ( x + y )^{2} - xy \\= (1) - (2) \\= 4 - 1 \\= 3$

Therefore.,

$\red { Value \:of \:x^{2} + y^{2} + xy} \green {= 3}$

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