Question

How do you factor ${x^2} - 9$ ?

Answer 1

Hint:
Here they have given ${x^2} - 9$ to find the factors, to make it simple for factorization try to represent $9$ in terms of squares of any number. Therefore we can rewrite the given expression as ${x^2} - {3^2}$ which represents the general form of difference of squares as ${a^2} - {b^2}$. Hence we can use ${a^2} - {b^2} = (a + b)(a - b)$ to find the factors of the given expression.

Complete step by step solution:
In the question they have given ${x^2} - 9$ to find the factors, to make it simple for factorization try to represent $9$ in terms of squares of any number.
Therefore we can rewrite the given expression as ${x^2} - {3^2}$ which represents the general form of difference of squares as ${a^2} - {b^2}$ . Hence we can use ${a^2} - {b^2} = (a + b)(a - b)$ to find the factors of the given expression.
Here in this question $a = x$ and $b = 3$ . By substituting these in the above formula, we get
${x^2} - {3^2} = \left( {x + 3} \right)\left( {x - 3} \right)$
The right hand side represents the factors of the given expression.
In order to simplify further we need to equate the right hand side expression that uses factors of the ${x^2} - {3^2}$ to zero.
Therefore, we get
$x + 3 = 0$ $x - 3 = 0$
On simplifying the above equation, we get
$ \Rightarrow x = - 3$
$ \Rightarrow x = 3$

Hence from the above calculation we can say that the two factors of the given expression ${x^2} - 9$ is $ - 3$ and $3$.

Note:
Whenever they ask to find the factors of a given expression, first try to represent the given expression in any of the general forms to simplify otherwise we will not get the correct answer. Try to remember the formulas for substitution. When finding the factors be careful with the sign that is positive and the negative sign.