Question

What should be added to \[{x^2} + xy + {y^2}\] to obtain \[2{x^2} + 3xy\] ?

Answer 1

Hint: We can find the required expression to be added by forming an equation using a variable and then solving that equation to find the value of the variable. As this variable represents the required equation, its value will be the required answer.

Complete step-by-step answer:
The given expressions are:
\[{x^2} + xy + {y^2}\]
\[2{x^2} + 3xy\]
We need to find an expression which will be added to the first to get the second expression. Let this required expression be exp. then according to the statement, we can develop a mathematical equation as:
\[{x^2} + xy + {y^2} + \exp = 2{x^2} + 3xy\]
Solving this equation to get the value of exp:
\[
   \Rightarrow \exp = 2{x^2} + 3xy - \left( {{x^2} + xy + {y^2}} \right) \\
   \Rightarrow \exp = 2{x^2} + 3xy - {x^2} - xy - {y^2} \\
   \Rightarrow \exp = 2{x^2} - {x^2} + 3xy - xy - {y^2} \\
   \Rightarrow \exp = {x^2} + 2xy - {y^2} \;
 \]
Therefore, we can add the expression \[{x^2} + 2xy - {y^2}\] to \[{x^2} + xy + {y^2}\] to obtain \[2{x^2} + 3xy\]
So, the correct answer is “ \[{x^2} + 2xy - {y^2}\] ”.

Note: While subtracting, remember that the like terms will be subtracted from their corresponding like terms. For example: we cannot subtract 3xy from $ 2{x^2} $ . The term containing ‘xy’ can be subtracted from another term containing ‘xy’.
Whenever an expression has a negative sign in front of it, the signs of all its term changes i.e. negative becomes positive and positive becomes negative.
This question could be interpreted in simple numbers without expressions like:
What will we add to 6 to get 10? We will simply subtract 6 from 10 to get the answer. Same goes for the expressions. To get the expression that should be added to the first to obtain the second, we just need the difference between the two.