Question

Evaluate the product of \[\left( 2a+3b \right)\left( 2a-3b \right)\]

Answer 1

Hint: We are given an algebraic expression and we are asked to evaluate the product in the expression. We will open one bracket and multiply the terms individually with the other brackets and so we get, \[2a\left( 2a-3b \right)+3b\left( 2a-3b \right)\]. Now, we will multiply \[2a\] with \[2a\] and \[-3b\]. Then, we will proceed to multiply \[3b\] with \[2a\] and \[-3b\]. We will have the values of these multiplications. And then we will cancel out terms which are the same but with opposite signs. And hence, we will have the required values of the given expression.

Complete step by step answer:
According to the given question, we are given an algebraic expression and we are asked to evaluate the product in the given expression.
The expression that we have is,
\[\left( 2a+3b \right)\left( 2a-3b \right)\]
We will now multiply each of the terms from one bracket to the terms in the other bracket, we get,
\[\Rightarrow 2a\left( 2a-3b \right)+3b\left( 2a-3b \right)\]
Now, we will multiply \[2a\] with \[2a\] and \[-3b\]. Then, we will proceed onto multiply \[3b\] with \[2a\] and \[-3b\], we now get,
\[\Rightarrow 4{{a}^{2}}-6ab+6ab-9{{b}^{2}}\]
As we can see that we have the value of the terms multiplied previously.
We will now cancel out the similar terms in the above expression with opposite signs and we get the new expression as,
\[\Rightarrow 4{{a}^{2}}-9{{b}^{2}}\]
Therefore, the value of the given expression is \[4{{a}^{2}}-9{{b}^{2}}\].

Note: The above expression can also be solved using the algebraic identity, which is as follows, \[\left( x+y \right)\left( x-y \right)={{x}^{2}}-{{y}^{2}}\]. Applying this in the given expression we get,
\[\left( 2a+3b \right)\left( 2a-3b \right)\]
\[\Rightarrow \left( {{\left( 2a \right)}^{2}}-{{\left( 3b \right)}^{2}} \right)\]
\[\Rightarrow 4{{a}^{2}}-9{{b}^{2}}\]
Therefore, we have the value of the given expression as, \[4{{a}^{2}}-9{{b}^{2}}\].