Certainly! Let's solve the expression [tex]\(25x^2 y - 121\)[/tex] step by step.
### Step 1: Recognize the Form of the Expression
The given expression is [tex]\(25x^2 y - 121\)[/tex]. Notice that this expression resembles the form of a difference of squares, which is commonly written as [tex]\(a^2 - b^2\)[/tex].
### Step 2: Identify the Squares
In a difference of squares expression [tex]\(a^2 - b^2\)[/tex], we have two perfect squares subtracted from each other. Our task is to identify values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] such that:
[tex]\[ a^2 = 25x^2 y \][/tex]
[tex]\[ b^2 = 121 \][/tex]
#### Breaking Down the Components:
- [tex]\(25x^2 y\)[/tex] can be viewed as [tex]\((5x\sqrt{y})^2\)[/tex]
- [tex]\(121\)[/tex] is a perfect square and can be written as [tex]\(11^2\)[/tex]
So, we can identify:
[tex]\[ a = 5x\sqrt{y} \][/tex]
[tex]\[ b = 11 \][/tex]
### Step 3: Apply the Difference of Squares Formula
The difference of squares formula states:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
Using our identified [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[ a = 5x\sqrt{y} \][/tex]
[tex]\[ b = 11 \][/tex]
We can rewrite the original expression [tex]\(25x^2 y - 121\)[/tex] as:
[tex]\[ (5x\sqrt{y})^2 - 11^2 \][/tex]
Applying the difference of squares formula:
[tex]\[ 25x^2 y - 121 = (5x\sqrt{y} - 11)(5x\sqrt{y} + 11) \][/tex]
### Conclusion
Thus, the factored form of the expression [tex]\(25x^2 y - 121\)[/tex] is:
[tex]\[ 25 x^2 y - 121 = (5x\sqrt{y} - 11)(5x\sqrt{y} + 11) \][/tex]
This completes the factorization of the given expression.
