Question

If x = 1
√2+√3
and y = 1
√2− √3
, find the value of x2 + xy + y2

Answer 1

Question :

$\sf \small \maltese \: if \: x = \frac{1}{ \sqrt{2} + \sqrt{3} } \: and \: y = \frac{1}{ \sqrt{2} - \sqrt{3} } \: then \: find \: the \: value \: of \: {x}^{2} + xy + {y}^{2}$

Solution :

First we need to rationalise the denominator in both x and y then simplify it according to the given condition

$\sf \small \dashrightarrow x = \frac{1}{ \sqrt{2} + \sqrt{3} }$

$\sf \small \dashrightarrow x = \frac{1( \sqrt{2} - \sqrt{3}) }{( \sqrt{2} + \sqrt{3} )( \sqrt{2} - \sqrt{3}) }$

$\sf \small \dashrightarrow x = \frac{ \sqrt{2} - \sqrt{3} }{ {( \sqrt{2} )}^{2} - {( \sqrt{3}) }^{2} }$

$\sf \small \dashrightarrow x = \frac{ \sqrt{2} - \sqrt{3} }{2 - 3}$

$\sf \small \dashrightarrow x = - \sqrt{2} + \sqrt{3}$

Now to rationalize y

$\sf \small \dashrightarrow y = \frac{1}{ \sqrt{2} - \sqrt{3} }$

$\sf \small \dashrightarrow y = \frac{1( \sqrt{2} + \sqrt{3}) }{( \sqrt{2} - \sqrt{3} )( \sqrt{2} + \sqrt{3}) }$

$\sf \small \dashrightarrow y= \frac{ \sqrt{2} + \sqrt{3} }{ {( \sqrt{2} )}^{2} - {( \sqrt{3}) }^{2} }$

$\sf \small \dashrightarrow y= \frac{ \sqrt{2} + \sqrt{3} }{2 - 3}$

$\sf \small \dashrightarrow y = - \sqrt{2} - \sqrt{3}$

Finally we rationalized both x and y

Now to simplify the given condition

$\sf \small \dashrightarrow {x}^{2} + xy + {y}^{2}$

Putting the values we get

$\sf \small \dashrightarrow ( - \sqrt{2} + \sqrt{3} )^{2} + ( - \sqrt{2} + \sqrt{3} )( - \sqrt{2} - \sqrt{3} ) + {( - \sqrt{2} - \sqrt{3} )}^{2}$

$\sf \tiny \dashrightarrow {( - \sqrt{2} )}^{2} + 2 ( - \sqrt{2} )( \sqrt{3} ) + {( \sqrt{3} )}^{2} + {( -\sqrt{2} )}^{2} - {( \sqrt{3}) }^{2} + ( - \sqrt{2} )^{2} - 2( - \sqrt{2} )( - \sqrt{3} ) + {( - \sqrt{3} )}^{2}$

$\sf \small \dashrightarrow 2 - 2 \sqrt{6} + 3 + 2 - 3 + 2 + 2 \sqrt{6} + 3$

$\sf \small \dashrightarrow 2 + 3 - \cancel{2 \sqrt{6}} + 2 + 3 + \cancel{2 \sqrt{6} } - 1$

$\sf \small \dashrightarrow 5 + 5 - 1$

$\sf \small \dashrightarrow 10 - 1$

$\sf \small \dashrightarrow 9$

So the value of x² + xy + y² is 9