To simplify the given expression [tex]\(\frac{x^2 - 4x + 4}{x^2 - 4}\)[/tex], follow these steps:
1. Factor the numerator and the denominator:
- The numerator [tex]\(x^2 - 4x + 4\)[/tex] is a quadratic expression that can be factored. Notice it's a perfect square:
[tex]\[
x^2 - 4x + 4 = (x - 2)^2
\][/tex]
- The denominator [tex]\(x^2 - 4\)[/tex] is a difference of squares. Rewrite it as:
[tex]\[
x^2 - 4 = (x - 2)(x + 2)
\][/tex]
2. Rewrite the expression using the factored forms:
[tex]\[
\frac{x^2 - 4x + 4}{x^2 - 4} = \frac{(x - 2)^2}{(x - 2)(x + 2)}
\][/tex]
3. Simplify the fraction:
- Notice that [tex]\((x - 2)\)[/tex] is a common factor in the numerator and the denominator. We can cancel one [tex]\((x - 2)\)[/tex] term from both the numerator and the denominator:
[tex]\[
\frac{(x - 2)^2}{(x - 2)(x + 2)} = \frac{x - 2}{x + 2}
\][/tex]
4. State the simplified expression:
The simplified form of the given expression is:
[tex]\[
\frac{x - 2}{x + 2}
\][/tex]
Hence, the expression [tex]\(\frac{x^2 - 4x + 4}{x^2 - 4}\)[/tex] simplifies to [tex]\(\frac{x - 2}{x + 2}\)[/tex].
