Question

How do you factor $2{\left( {x - 2y} \right)^2} - 4x\left( {x - 2y} \right)$?

Answer 1

Hint: We will first take the factor $\left( {x - 2y} \right)$ common from the expression $2{\left( {x - 2y} \right)^2} - 4x\left( {x - 2y} \right)$. Then we will just simplify the inside expression further to get the answer.

Complete step by step solution:
We are given that we are required to factor $2{\left( {x - 2y} \right)^2} - 4x\left( {x - 2y} \right)$.
Since, we see that both the terms have $\left( {x - 2y} \right)$ as the common factor. Therefore, taking $\left( {x - 2y} \right)$ common from both the terms, we will then obtain the following expression with us:-
$ \Rightarrow 2{\left( {x - 2y} \right)^2} - 4x\left( {x - 2y} \right) = \left( {x - 2y} \right)\left\{ {2\left( {x - 2y} \right) - 4x} \right\}$
Simplifying the terms inside the curly bracket on the right hand side of the above expression, we will then obtain the following expression with us:-
$ \Rightarrow 2{\left( {x - 2y} \right)^2} - 4x\left( {x - 2y} \right) = \left( {x - 2y} \right)\left\{ {2x - 4y - 4x} \right\}$
Simplifying the terms inside the curly bracket on the right hand side of the above expression by clubbing the like terms, we will then obtain the following expression with us:-
$ \Rightarrow 2{\left( {x - 2y} \right)^2} - 4x\left( {x - 2y} \right) = \left( {x - 2y} \right)\left( { - 2x - 4y} \right)$
Taking the negative sign common from the latter bracket, we will then obtain the following expression with us:-
$ \Rightarrow 2{\left( {x - 2y} \right)^2} - 4x\left( {x - 2y} \right) = - \left( {x - 2y} \right)\left( {2x + 4y} \right)$
Taking the negative sign common inside the first bracket, we will then obtain the following expression with us:-
$ \Rightarrow 2{\left( {x - 2y} \right)^2} - 4x\left( {x - 2y} \right) = \left( {2y - x} \right)\left( {2x + 4y} \right)$


Note: The students must note that they might make the mistake of already opening up the square of $\left( {x - 2y} \right)$ but that might make the given question difficult and it will be then difficult for us to club the terms and find common terms. Therefore, just take out the factor of $\left( {x - 2y} \right)$ common already from the given expression without simplifying it from the beginning.
Now, the students must also note the underlying distributive property has been used in the above solution.
It states that a (b + c) = ab + ac for any numbers a, b and c.
Here, we just replaced a by 2, b by x and c by – 2y, we will then obtain the following equation with us:-
$ \Rightarrow 2\left( {x - 2y} \right) = 2x - 4y$