To factor the quadratic expression [tex]\( x^2 - 10x + 25 \)[/tex] completely, let's follow these steps:
1. Identify the quadratic expression: [tex]\( x^2 - 10x + 25 \)[/tex].
2. Recognize it as a perfect square trinomial: A perfect square trinomial takes the form [tex]\( a^2 - 2ab + b^2 \)[/tex]. In this expression, if we identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex], we can rewrite it as a square of a binomial.
3. Identify the terms [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
- The coefficient of [tex]\(x^2\)[/tex] is [tex]\(1\)[/tex], so [tex]\( a = x \)[/tex].
- The constant term is [tex]\(25\)[/tex], and [tex]\(25\)[/tex] is a perfect square because [tex]\( 25 = 5^2 \)[/tex]. Hence, [tex]\( b = 5 \)[/tex].
4. Verify the middle term:
- The middle term [tex]\(-10x\)[/tex] needs to be [tex]\(-2ab\)[/tex]. In this case, [tex]\(-2 \cdot x \cdot 5 = -10x \)[/tex], which matches our expression.
5. Write the expression as a square of a binomial:
- Putting it all together, we have:
[tex]\[ x^2 - 10x + 25 = (x - 5)^2 \][/tex]
6. Express the factored form:
- Therefore, the expression [tex]\( x^2 - 10x + 25 \)[/tex] can be factored completely as [tex]\( (x - 5)^2 \)[/tex].
Thus, the correct factorization is [tex]\((x - 5)(x - 5)\)[/tex], which can also be written as [tex]\((x - 5)^2\)[/tex]. This matches the first option:
[tex]\[
(x-5)(x-5)
\][/tex]
