Question

How do you multiply \[{\left( {2a + 3b} \right)^2}\]?

Answer 1

Hint: This problem deals with expanding algebraic expressions. While expanding an algebraic expression, we combine more than one number or variable by performing the given algebraic operations. The basic steps to simplify an algebraic expression are: remove parentheses by multiplying factors, use exponent rules to remove parentheses in terms with exponents, combine like terms by adding coefficients, and then combine the constants.

Complete step-by-step solution:
First we split the given expression as :
$ \Rightarrow {\left( {2a + 3b} \right)^2} = \left( {2a + 3b} \right)\left( {2a + 3b} \right)$
Given a product of two linear expressions in$a$ and $b$, and we have to expand the given expression and simplify the expression, as given below:
\[ \Rightarrow \left( {2a + 3b} \right)\left( {2a + 3b} \right)\]
Now expanding the given expression by using the distributive property.
Consider the given linear expression, as given below:
\[ \Rightarrow 2a\left( {2a + 3b} \right) + 3b\left( {2a + 3b} \right)\]
Now using the distributive property, multiplying the number with the all the terms of the expression in the bracket, as shown below:
\[ \Rightarrow \left( {2a} \right)2a + \left( {2a} \right)3b + \left( {3b} \right)2a + \left( {3b} \right)3b\]
Now after the expansion of the multiplication of the terms, simplifying the terms as given below:
\[ \Rightarrow 4{a^2} + 6ab + 6ab + 9{b^2}\]
Now grouping and simplifying the like terms and unlike terms as shown below:
\[ \Rightarrow 4{a^2} + 12ab + 9{b^2}\]
$\therefore {\left( {2a + 3b} \right)^2} = 4{a^2} + 12ab + 9{b^2}$

The value of the expression on expanding \[{\left( {2a + 3b} \right)^2}\] is equal to $4{a^2} + 12ab + 9{b^2}$.

Note: While solving this problem please note that we expanded the given expression with the help of distributive property. This is done by using the distributive property to remove any parentheses or brackets and by combining the like terms and unlike terms. If you see parenthesis with more than one term inside, then distribute first. This problem can also be done by the formula of ${\left( {x + y} \right)^2} = {x^2} + 2xy + {y^2}$.