Answer :

To factor the quadratic expression [tex]\( s^2 - 4s + 4 \)[/tex] completely, follow these steps:

1. Identify the structure of a perfect square trinomial:

A perfect square trinomial can be written in the form [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex].

2. Compare the given expression to this structure:

The expression is [tex]\( s^2 - 4s + 4 \)[/tex].
- Here, [tex]\( a = s \)[/tex].
- Notice that [tex]\( a^2 = s^2 \)[/tex].
- The last term is 4, which is a perfect square, [tex]\( b^2 = 4\)[/tex] so [tex]\( b = 2 \)[/tex].
- The middle term is [tex]\(-4s\)[/tex], which should equal [tex]\( -2 \times a \times b \)[/tex].

3. Verify the middle term calculation:

- Calculate [tex]\( 2ab = 2 \times s \times 2 = 4s \)[/tex].
- The expression [tex]\(-4s\)[/tex] matches the middle term when the expression is expanded.

4. Write the expression as a square:

Since all conditions are satisfied, the expression can be factored as:
[tex]\[
(s - 2)^2
\][/tex]

5. Conclusion:

The completely factored form of [tex]\( s^2 - 4s + 4 \)[/tex] is [tex]\((s - 2)(s - 2)\)[/tex] or simply [tex]\((s - 2)^2\)[/tex].

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