Sure, I'd be happy to help you factor the expression [tex]\( 169x^2 - 4 \)[/tex]. Here is the step-by-step solution:
1. Identify the general form:
The expression [tex]\( 169x^2 - 4 \)[/tex] is a quadratic expression in the form of [tex]\( ax^2 + bx + c \)[/tex]. In this case:
- [tex]\( a = 169 \)[/tex]
- [tex]\( b = 0 \)[/tex]
- [tex]\( c = -4 \)[/tex]
2. Recognize it as a difference of squares:
Notice that this expression can be written as a difference of squares. A difference of squares is an expression of the form [tex]\( A^2 - B^2 \)[/tex] which factors into [tex]\( (A - B)(A + B) \)[/tex].
3. Rewrite the terms as squares:
We can rewrite both terms of the expression as squares:
- [tex]\( 169x^2 \)[/tex] is [tex]\((13x)^2\)[/tex]
- [tex]\( 4 \)[/tex] is [tex]\( (2)^2 \)[/tex]
So, the expression [tex]\( 169x^2 - 4 \)[/tex] can be seen as [tex]\( (13x)^2 - (2)^2 \)[/tex].
4. Apply the difference of squares formula:
The difference of squares formula states that [tex]\( A^2 - B^2 = (A - B)(A + B) \)[/tex]. Here:
- [tex]\( A = 13x \)[/tex]
- [tex]\( B = 2 \)[/tex]
So, we substitute [tex]\( A \)[/tex] and [tex]\( B \)[/tex] into the formula to get:
[tex]\[
(13x)^2 - (2)^2 = (13x - 2)(13x + 2)
\][/tex]
5. Write the final factored form:
Therefore, the factored form of [tex]\( 169x^2 - 4 \)[/tex] is:
[tex]\[
(13x - 2)(13x + 2)
\][/tex]
So, the expression [tex]\( 169x^2 - 4 \)[/tex] factors to [tex]\( (13x - 2)(13x + 2) \)[/tex].
