Question

If x2 + xy + y = 84 & x - / xy + y = 6 , then find x + y . यदि x2 + xy + y = 84 और x - / xy + y = 6 है , तबx y का मान क्या होगा ?

Answer 1

Answer:

Required value of x^3 + y^3 is 520.

Step-by-step explanation:

Given,

x^2 + xy + y^2 = 84 ...( 1 )

Also, x - √xy + y = 6

= > x - √xy + y = 6

= > x + y = 6 + √xy

= > ( x + y )^2 = ( 6 + √xy )^2

= > x^2 + y^2 + 2xy = 36 + xy + 12√xy

= > x^2 + y^2 + 2xy - xy = 36 + 12√xy

= > x^2 + y^2 + xy = 36 + 12√xy

= > 84 = 36 + 12√xy { from ( 1 ) }

= > 48 = 12√xy

= > 4 = √xy

Thus,

= > x - √xy + y = 6

= > x - 4 + y = 6

= > x + y = 10 ...( 2 )

Substituting the value of√xy in ( 1 ) :

= > x^2 + xy + y^2 = 84

= > x^2 + 2xy - xy + y^2 = 84

= > x^2 - xy + y^2 + 2xy = 84

= > x^2 - xy + y^2 + 2( 4 )^2 = 84

= > x^2 - xy + y^2 + 32 = 84

= > x^2 - xy + y^2 = 52 ...( 3 )

We know,

a^3 + b^2 = ( a + b )( a^2 - ab + b^2 )

Therefore,

= > x^3 + y^3

= > ( x + y )( x^2 - xy + y^2 )

= > ( 10 )( 52 ) { from ( 2 ) and ( 3 ) }

= > 520

Hence the required value of x^3 + y^3 is 520.

Answer 2

$\Large{\textbf{\underline{\underline{According\:to\:the\:Question}}}}$

Information is given that :-

x² + xy + y² = 84 ........... (1)

Here,

x - √xy + y = 6

x - √xy + y = 6

x + y = 6 + √xy

(x + y)² = (6 + √xy)²

x² + y² + 2xy = 36 + xy + 12√xy

x² + y² + 2xy - xy = 36 + 12√xy

x² + y² + xy = 36 + 12√xy

From (1) we have,

84 = 36 + 12√xy

48 = 12√xy

4 = √xy

Thus,

x - √xy + y = 6

x - 4 + y = 6

x + y = 10 ............. (2)

Putting value of√xy in (1) :

x² + xy + y² = 84

x² + 2xy - xy + y² = 84

x² - xy + y² + 2xy = 84

x² - xy + y² + 2(4)² = 84

x² - xy + y² + 32 = 84

x² - xy + y² = 52 .......... (3)

$\Large{\boxed{\sf\:{a^3+b^3=(a+b)(a^2-ab+b^2)}}}$

Therefore,

= x³ + y³= (x + y)(x² - xy + y²)

From (2) and (3)

= 10 × 52

= 520

Therefore,

Required value is 520