Question

How do you factor completely $16{x^2} - 25{y^2}$?

Answer 1

Hint: We will first of all use the formula ${a^2} - {b^2} = (a - b)(a + b)$ and replace a and b by the required things so that we get the two factors of the given expression.

Complete step-by-step answer:
We are given that we are required to factorize $16{x^2} - 25{y^2}$.
We know that we have an identity given by the following expression:-
$ \Rightarrow {a^2} - {b^2} = (a - b)(a + b)$
Now, replace a by 4x and b by 5y in the above written identity to get the following expression with us:-
$ \Rightarrow {\left( {4x} \right)^2} - {\left( {5y} \right)^2} = (4x - 5y)(4x + 5y)$
Now, if we simplify the squaring on the left hand side, we will then obtain the following expression:-
$ \Rightarrow 16{x^2} - 25{y^2} = (4x - 5y)(4x + 5y)$

Hence, the factors of $16{x^2} - 25{y^2}$ are (4x – 5y) and (4x + 5y).

Note:
The students must note that there is an alternate way to do the same and get the same factors. In that method, we will add and subtract something similar in the given equation so that we can take things common and make the similar factors. This can be done as follows:-
We are given that we are required to factorize $16{x^2} - 25{y^2}$.
Let us add and subtract 20 xy to it. Then, we will obtain the following expression:-
$ \Rightarrow 16{x^2} - 25{y^2} - 20xy + 20xy$
Now, we can re - arrange the terms and write the expression in the above line as the following:-
$ \Rightarrow 16{x^2} - 20xy - 25{y^2} + 20xy$
Now, we will use associative property and combine the terms as pairs in the above given expression to obtain the expression written below:-
$ \Rightarrow \left( {16{x^2} - 20xy} \right) + \left( { - 25{y^2} + 20xy} \right)$
Now, we can take 5x common among the terms $16{x^2}$ and 20 xy given in front and 5y common from $25{y^2}$ and 20 xy given in back, then we will thus obtain the following expression:-
$ \Rightarrow 4x\left( {4x - 5y} \right) + 5y\left( { - 5y + 4x} \right)$
Since we have similar terms in bracket as well, we can take it common to get the following expression:-
$ \Rightarrow (4x - 5y)(4x + 5y)$
Thus we have the required factors.