To factor the expression [tex]\( 49x^2 - 121 \)[/tex] completely, follow these steps:
1. Recognize the Form: Notice that the given expression is a difference of squares. The difference of squares is a special algebraic form which can be expressed as [tex]\( a^2 - b^2 \)[/tex].
2. Rewrite the Expression: We need to express [tex]\( 49x^2 - 121 \)[/tex] in the form [tex]\( a^2 - b^2 \)[/tex].
- Identify [tex]\( a \)[/tex] and [tex]\( b \)[/tex] such that [tex]\( a^2 = 49x^2 \)[/tex] and [tex]\( b^2 = 121 \)[/tex].
- Here, [tex]\( 49x^2 \)[/tex] can be written as [tex]\( (7x)^2 \)[/tex], so [tex]\( a = 7x \)[/tex].
- Similarly, [tex]\( 121 \)[/tex] can be written as [tex]\( 11^2 \)[/tex], so [tex]\( b = 11 \)[/tex].
3. Apply the Difference of Squares Formula: The formula for the difference of squares is:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
Using the identified values [tex]\( a = 7x \)[/tex] and [tex]\( b = 11 \)[/tex]:
[tex]\[
49x^2 - 121 = (7x)^2 - 11^2 = (7x - 11)(7x + 11)
\][/tex]
4. Write the Factored Form: Putting it all together, the completely factored form of the expression [tex]\( 49x^2 - 121 \)[/tex] is:
[tex]\[
49x^2 - 121 = (7x - 11)(7x + 11)
\][/tex]
Therefore, the factors of [tex]\( 49x^2 - 121 \)[/tex] are [tex]\( (7x - 11) \)[/tex] and [tex]\( (7x + 11) \)[/tex].
