Question

How do you simplify \[{\left( {3x + 2y} \right)^2}\]?

Answer 1

Hint:
Given is a bracket with two terms $\left( {3x} \right)$ and $\left( {2y} \right)$. Given is the square of the term. Square is nothing but multiplying the same number with itself. So we will multiply the bracket with itself. Then each individual term in the first bracket is multiplied with that in the second term. Then if needed any mathematical operations, those will be performed. Or else we can use the standard and important identities used for expansion like those used in squaring or cubing. Such identities are known as algebraic identities.

Complete step by step solution:
Given that
\[\left( {3x + 2y} \right)\]
Now taking square of the term
\[ \Rightarrow \left( {3x + 2y} \right)\left( {3x + 2y} \right)\]
First take $\left( {3x} \right)$ and multiply it with both the terms of the second term or same can be done for $\left( {2y} \right)$ with the terms from the first bracket,
\[ \Rightarrow \left( {3x} \right)\left( {3x + 2y} \right) + \left( {2y} \right)\left( {3x + 2y} \right)\]
\[ \Rightarrow \left( {3x \times 3x} \right) + \left( {3x \times 2y} \right) + \left( {2y \times 3x} \right) + \left( {2y \times 2y} \right)\]
\[ \Rightarrow \left( {9{x^2}} \right) + \left( {6xy} \right) + \left( {6xy} \right) + \left( {4{y^2}} \right)\]
Now simplify the expressions, we get,
\[ \Rightarrow 9{x^2} + 12xy + 4{y^2}\]
This is the simplified answer of the given problem.

Note:
In this problem, we were required to evaluate the square of the algebraic expression given to us. In the process of doing the required task, we have to simplify the product by adding the like terms. In a similar way, if we are required to evaluate a cube of an expression, then we multiply the same term three times. There are various important identities that help us in finding the square and cube expansion of the same problem directly such as the one used in the given question itself.