If $2x+y=12, xy=10$ then find the value of $8x^3 +y^3$

Given: $2 x + y = 12 ; x y  =  10$

To find: The value of $8x^3 +y^3$  

Solution: 

Cubing on both sides of  $2 x + y = 12$

$(2 x + y)^3 = 12 ^3$  

$(a+b)^3= a^3  + b^3  + 3 a b (a+b)$

 $(2 x + y)^3  =  (2 x)^3  + y^3   +  3 \times 2 x \times y (2 x + y)$

$(2 x + y)^3 =  8 x^3  + y^3   +  6 x y (2 x + y )$

$(2 x + y)^3 =  8 x^3  + y^3   + 6 \times10 \times 12$

$(2 x + y)^3  =  8 x ^3 + y ^3   + 720$

$2 x + y = 12 ; 12 ^3 = 1728$

$1728 = 8 x^3  + y ^3   + 720$

$1728 - 720 = 8 x^3  + y^3$   

$1008 = 8 x^3  + y^3$   

Rewrite,

$8 x^3  + y ^3  = 1008$

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Updated on: 10-Oct-2022

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