Question

If x=√2+1/√2-1 and y=√2-1/√2+1, find x²+y²+xy.​

Answer 1

${\large{\underline{\underline{\pmb{\frak{Let's \:understand\:the\:concept :-}}}}}}$

As per the given information, we have the values of x and y, We need to rationalize the values, By Multiplying the values with its rationalising factors, After rationalising the values of x and y, we can find the the value of the given equation i.e x² + y²+ xy. by putting the rationalized values in the equation.

${\large{\underline{\underline{\pmb{\frak{Given:-}}}}}}$

$\sf \bullet \;\; x = \dfrac{ \sqrt{2 } + 1 }{ \sqrt{2} - 1 } \: , \: y=\dfrac{ \sqrt{2 } - 1 }{ \sqrt{2} + 1 },$

${\large{\underline{\underline{\pmb{\frak{solution:-}}}}}}$

Rationalising the value of x,

$\sf : \;\implies \dfrac{ \sqrt{2} + 1}{ \sqrt{2} - 1}$

$\sf : \;\implies \dfrac{ \sqrt{2} + 1}{ \sqrt{2} - 1} \times \dfrac{ \sqrt{2} + 1}{ \sqrt{2} + 1}$

$\sf : \;\implies \dfrac{ ({\sqrt{2} + 1)}^{2} }{ ({2 - 1})}$

$\sf : \;\implies \dfrac{( {\sqrt{2} + 1})^{2} }{ {1}}$

$\sf : \;\implies ( {\sqrt{2} + 1})^{2}$

$\sf : \;\implies {( \sqrt{2} )}^{2} + {(1)}^{2} + 2( \sqrt{2} )(1)$

$\sf : \;\implies 2 + 1 +2 \sqrt{2}$

$\sf : \;\implies 3 +2 \sqrt{2}$

Now, Rationalising the value of y,

$\sf : \;\implies \dfrac{ \sqrt{2 } - 1 }{ \sqrt{2} + 1 }$

$\sf : \;\implies \dfrac{ \sqrt{2 } - 1 }{ \sqrt{2} + 1 } \times \dfrac{ \sqrt{2 } - 1 }{ \sqrt{2} - 1 }$

$\sf : \;\implies \dfrac{ ( {\sqrt{2 } - 1 )}^{2}}{ {2} - 1 }$

$\sf : \;\implies ( {{\sqrt{2} - 1})^{2} }$

$\sf : \;\implies {( \sqrt{2} )}^{2} + {(1)}^{2} - 2( \sqrt{2} )(1)$

$\sf : \;\implies 2 + 1 - 2 \sqrt{2}$

$\sf : \;\implies 3 - 2 \sqrt{2}$

Now, solving the equation by putting the values,

$: \implies \sf {x}^{2} + {y}^{2} +xy$

${ : \implies \sf {(3 + 2 \sqrt{2} )}^{2} + {(3 - 2 \sqrt{2} )}^{2} +(3 + 2 \sqrt{2} ) (3 - 2 \sqrt{2} )}$

${ : \implies \sf 17 + 12\sqrt{2} + {(3 - 2 \sqrt{2} )}^{2} +(3 + 2 \sqrt{2} ) (3 - 2 \sqrt{2} )}$

${ : \implies \sf 17 + 12\sqrt{2} + 17 - 12\sqrt{2} +(3 + 2 \sqrt{2} ) (3 - 2 \sqrt{2} )}$

On cancelling 12√2 & -12√2,

${ : \implies \sf 17 + 17 +(3 + 2 \sqrt{2} ) (3 - 2 \sqrt{2} )}$

${ : \implies \sf 34 +(3 + 2 \sqrt{2} ) (3 - 2 \sqrt{2} )}$

${ : \implies \sf 34 + 1}$

$\boxed{\bf{ \leadsto{ 35 }}}$