Question

Expand \[\left( {2x + 3y} \right)\left( {2x - 3y} \right)\].

Answer 1

Hint: Here, we will use the algebraic identity for the product of the sum and difference of two numbers. Then, we will simplify the expression to get the required expansion of \[\left( {2x + 3y} \right)\left( {2x - 3y} \right)\].

Formula Used: We will use the formula of the product of the sum and difference of two numbers \[a\] and \[b\] is given by the algebraic identity \[\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}\].

Complete step-by-step answer:
We need to find the product of the sum and difference of the numbers \[2x\] and \[3y\].
Substituting \[a = 2x\] and \[b = 3y\] in the algebraic identity, we get
\[ \Rightarrow \left( {2x + 3y} \right)\left( {2x - 3y} \right) = {\left( {2x} \right)^2} - {\left( {3y} \right)^2}\]
We will simplify the expression on the right hand side.
Applying the exponents on the bases, we get
\[ \Rightarrow \left( {2x + 3y} \right)\left( {2x - 3y} \right) = 4{x^2} - 9{y^2}\]
Since there are no like terms, we cannot simplify the expression further.
\[\therefore \] The expansion of \[\left( {2x + 3y} \right)\left( {2x - 3y} \right)\] is \[4{x^2} - 9{y^2}\].

Note: We cannot simplify \[4{x^2} - 9{y^2}\] further because there are no like terms in the expression. Like terms are the terms whose variables and their exponents are the same. For example, \[100x,150x,240x,600x\] all have the variable \[x\] raised to the exponent 1. Terms which are not like, and have different variables, or different degrees of variables cannot be added together. For example, it is not possible to add \[4{x^2}\] to \[ - 9{y^2}\].
We can also use the distributive law of multiplication to expand the expression. The distributive law of multiplication states that \[\left( {a + b} \right)\left( {c + d} \right) = a \cdot c + a \cdot d + b \cdot c + b \cdot d\].
Simplifying the expression \[\left( {2x + 3y} \right)\left( {2x - 3y} \right)\] using the distributive law of multiplication, we get
\[ \Rightarrow \left( {2x + 3y} \right)\left( {2x - 3y} \right) = \left( {2x} \right)\left( {2x} \right) + \left( {2x} \right)\left( { - 3y} \right) + \left( {3y} \right)\left( {2x} \right) + \left( {3y} \right)\left( { - 3y} \right)\]
Multiplying the terms, we get
\[ \Rightarrow \left( {2x + 3y} \right)\left( {2x - 3y} \right) = 4{x^2} - 6xy + 6xy - 9{y^2}\]
Adding and subtracting the like terms, we get
\[ \Rightarrow \left( {2x + 3y} \right)\left( {2x - 3y} \right) = 4{x^2} - 9{y^2}\]
\[\therefore \] The expansion of \[\left( {2x + 3y} \right)\left( {2x - 3y} \right)\] is \[4{x^2} - 9{y^2}\].