Question

pls answer with steps . show the steps properly pls ​

Answer 1

Answer:

Required value of x^2 - y^2 is ( 28 / 11 ) ( - 10√3 / 11 ) .

Step-by-step explanation:

It is given that the value of x is $\dfrac{5-\sqrt{3}}{5+\sqrt{3}}$ and the value of y is $\dfrac{5+\sqrt{3}}{5-\sqrt{3}}$ .

By Rationalization : -

$\implies x=\dfrac{5-\sqrt{3}}{5+\sqrt{3}} \times\dfrac{5-\sqrt{3}}{5-\sqrt{3}}\\\\\\\implies x = \dfrac{(5-\sqrt{3})^2}{(5+\sqrt{3})(5-\sqrt{3}}\\\\\\\implies x=\dfrac{25 + 3 -10\sqrt3 }{(5)^2 -(\sqrt{3})^2} \quad\quad \quad \mathsf{\bigg|(a-b)^2=a^2+b^2-2ab} ; (a+b)(a-b)=a^2-b^2\\\\\\\implies x= \dfrac{28-10\sqrt3}{22}$

Similarly,

$\implies y=\dfrac{5+\sqrt{3}}{5-\sqrt{3}} \times\dfrac{5+\sqrt{3}}{5+\sqrt{3}}\\\\\\\implies y = \dfrac{(5+\sqrt{3})^2}{(5-\sqrt{3})(5+\sqrt{3}}\\\\\\\implies y=\dfrac{25 + 3 +10\sqrt3 }{(5)^2 -(\sqrt{3})^2} \quad\quad \quad \mathsf{\bigg|(a+b)^2=a^2+b^2+2ab} ; (a+b)(a-b)=a^2-b^2\\\\\\\implies y= \dfrac{28+10\sqrt3}{22}$

We know ( a - b )( a + b ) = a^2 - b^2, thus x^2 - y^2 = ( x + y )( x - y )

Hence,

= > x^2 - y^2

= > ( x + y )( x - y )

= > { ( 28 - 10√3 ) / 22 + ( 28 + 10√3 ) / 22 }{ ( 28 - 10√3 ) / 22 - ( 28 + 10√3 ) / 22 }

= > { ( 28 + 10√3 + 28 - 10√3 ) / 22 } { ( 28 - 10√3 - 28 - 10√3 ) / 22 }

= > { 56 / 22 } { - 20√3 / 22 }

= > { 28 / 11 } { - 10√3 / 11 }

Hence the required value of x^2 - y^2 is ( 28 / 11 ) ( - 10√3 / 11 ) .

Your question needs a correction, question is ( x - y ) : -

= > ( x - y )

= > ( 28 - 10√3 ) / 22 - ( 28 + 10√3 ) / 22

= > ( 28 - 10√3 - 28 - 10√3 ) / 22  

= >  - 20√3 / 22

= >  - 10√3 / 11 .