To solve the problem of factoring [tex]\( x^2 - 9 \)[/tex], let's use the difference of squares formula. Here is a step-by-step breakdown of the process:
1. Identify the Given Expression:
We are given the expression [tex]\( x^2 - 9 \)[/tex].
2. Recognize the Form:
Notice that [tex]\( x^2 - 9 \)[/tex] is in the form of a difference of squares. The general formula for the difference of squares is:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are any expressions.
3. Rewrite the Given Expression to Match the Formula:
In our expression [tex]\( x^2 - 9 \)[/tex]:
- [tex]\( a \)[/tex] is [tex]\( x \)[/tex]
- [tex]\( b \)[/tex] is [tex]\( 3 \)[/tex]
Thus, [tex]\( x^2 - 9 \)[/tex] can be thought of as [tex]\( x^2 - 3^2 \)[/tex].
4. Apply the Difference of Squares Formula:
Substitute [tex]\( a = x \)[/tex] and [tex]\( b = 3 \)[/tex] into the difference of squares formula:
[tex]\[
x^2 - 3^2 = (x - 3)(x + 3)
\][/tex]
5. Write the Factored Form:
Therefore, the factored form of the expression [tex]\( x^2 - 9 \)[/tex] is:
[tex]\[
(x - 3)(x + 3)
\][/tex]
So, the expression [tex]\( x^2 - 9 \)[/tex] factors to [tex]\( (x - 3)(x + 3) \)[/tex].
