Question

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

Answer 1

Hint: Given is a bracket with two terms \[3x\] and \[2y\] . 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. Those are the algebraic identities.

Complete step-by-step answer:
Given that
 \[{\left( {3x - 2y} \right)^2}\]
Now taking square of the term
 \[ \Rightarrow \left( {3x - 2y} \right)\left( {3x - 2y} \right)\]
First take \[3x\] and multiply it with both the terms of the second term and same for 2y from first bracket,
 \[ \Rightarrow 3x\left( {3x - 2y} \right) - 2y\left( {3x - 2y} \right)\]
 \[ \Rightarrow (3x \times 3x) - (3x \times 2y) - (2y \times 3x) + (2y \times 2y) \]
Now simplify these terms,
 \[ \Rightarrow 9{x^2} - 6xy - 6yx + 4{y^2}\]
But middle two terms are having same variables \[xy\] so they can be added,
 \[ \Rightarrow 9{x^2} - 12yx + 4{y^2}\]
This is the simplified answer of the given problem.
So, the correct answer is “$9{x^2} - 12yx + 4{y^2}$”.

Note: In this problem we are taking the square of the term. After that when we are simplifying the product we have added the terms with the same coefficient we can say. If we are asked to find the cube of the term then we take the same term three times. The answer so obtained is a trinomial with highest degree 2.
There are various important identities that help us in finding the square and cube expansion of the same problem directly. How? Let’s see.
 \[{\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}\] this can be used directly to find the square.