To factorize the given quadratic expression [tex]\(x^2 - 4x + 4\)[/tex], let's proceed step-by-step:
1. Write down the quadratic expression:
[tex]\[ x^2 - 4x + 4 \][/tex]
2. Recognize the form:
Notice that this is a quadratic expression of the form [tex]\(ax^2 + bx + c\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = -4\)[/tex], and [tex]\(c = 4\)[/tex].
3. Identify perfect square trinomial:
Observe that the expression looks like it might be a perfect square trinomial because of the square terms and the middle term. A perfect square trinomial has the general form [tex]\((x - a)^2 = x^2 - 2ax + a^2\)[/tex].
4. Compare with the form [tex]\( (x-a)^2 \)[/tex]:
Compare [tex]\( x^2 - 4x + 4 \)[/tex] to [tex]\( (x - a)^2 = x^2 - 2ax + a^2 \)[/tex]:
- Here, [tex]\(a = 2\)[/tex] because the middle term [tex]\(-4x\)[/tex] can be written as [tex]\(-2 \cdot 2 \cdot x\)[/tex].
5. Check the constant term:
The constant term [tex]\(4\)[/tex] can be written as [tex]\(2^2\)[/tex].
6. Rewrite the quadratic expression:
Given the observations above, we can rewrite [tex]\( x^2 - 4x + 4 \)[/tex] as:
[tex]\[ (x - 2)^2 \][/tex]
7. Verify by expanding:
To confirm the factorization, expand [tex]\((x - 2)^2\)[/tex]:
[tex]\[ (x - 2)(x - 2) = x^2 - 2x - 2x + 4 = x^2 - 4x + 4 \][/tex]
Since the expanded form matches the original quadratic expression, the factorized form is correct.
Conclusion:
The correct factorized form of [tex]\( x^2 - 4x + 4 \)[/tex] is [tex]\((x - 2)^2\)[/tex].
Thus, the correct choice is:
[tex]\[ \boxed{(x - 2)^2} \][/tex]
