Question

How do you factor $4{x^2} - 81?$

Answer 1

Firstly check for any common factor between the terms of the given expression, if found then take the common factor out and if not found any common factor then try to write the expression in such a way that it will fit the following algebraic identity:
${a^2} - {b^2} = (a + b)(a - b)$ And use this identity to simply factorize the given equation.

Complete step by step solution:
In first we will check for any common factor between the terms of the expression $4{x^2} - 81$, which can be done as listing the factors of each term, that is

$4{x^2} = 4 \times x \times x \\\ 81 = 3 \times 3 \times 3 \times 3 \\\$
We can see that there is nothing common in both the terms, but do not worry finding common factors is not the only method for factoring an expression,
We have an algebraic identity which is written as follows
${a^2} - {b^2} = (a + b)(a - b)$
So let us try if we could fit the given expression $4{x^2} - 81$ in this identity or not
$= 4{x^2} - 81$
Trying to make the terms as square of particular number,
So in order to make the terms of the expression $4{x^2} - 81$ as square of particular number, we can write $4{x^2} = {(2x)^2}$ and also $81 = {9^2}$

Now putting the particular numbers in above identity in order to factorize $4{x^2} - 81$, we will get
$= 4{x^2} - 81 \\\ = {(2x)^2} - {9^2} \\\$
We can further write it as
$= {(2x)^2} - {9^2} \\\ = (2x + 9)(2x - 9) \\\$
$\therefore$ the required factored form of $4{x^2} - 81$ is $(2x + 9)(2x - 9)$

Note: Algebraic identities are very helpful in order to solve many algebraic questions even this ${a^2} - {b^2} = (a + b)(a - b)$ algebraic identity is useful for solving even trigonometric questions. Identities are something that can never be changed. They make the solution easier and help simplify a question in an easy and short way.