Question

what should be added to
${x}^{2} + xy + {y}^{2}$
to obtain
$2 {x}^{2} + 3xy$
???​

Answer 1

$\bold { \red{CLARIFICATION:}}$

Question :

$\tt What \: should \: be \: added \: to \: {x}^{2} + xy + {y}^{2}$

$\tt to \: obtain \: 2 {x}^{2} + 3xy$

Solution :

Let 'a' be added to the equation

$\tt {x}^{2} + xy + {y}^{2} + a = 2 {x}^{2} + 3xy$

$\implies \tt a = 2 {x}^{2} + 3xy - {x}^{2} - xy - {y}^{2}$

Bringing like terms together and cancelling the values,

$\implies \tt a = 2 {x}^{2} - {x}^{2} + 3xy - xy - {y}^{2}$

$\implies \tt a = {x}^{2}+ 2xy - {y}^{2}$

Therefore,

$\tt {x}^{2}+ 2xy - {y}^{2} \: \: Should \: be \: added \: to \: {x}^{2} + xy + {y}^{2}$

To get

$\tt2 {x}^{2} + 3xy$

$\bold{ \red{VERIFICATION:}}$

Let's add,

$\implies \tt ({x}^{2} + xy + {y}^{2}) + ({x}^{2}+ 2xy - {y}^{2})$

Bringing similar terms together,

$\implies \tt {x}^{2} + {x}^{2}+ 2xy + xy - {y}^{2}+ {y}^{2}$

We get ,

$\implies \tt2 {x}^{2} + 3xy$

Hence proved

$\tt {x}^{2}+ 2xy - {y}^{2} \: \: Should \: be \: added \: to \: the \: given \: equation$