To complete the given equation:
[tex]\[
x^2 + 10x + \ldots = (\ldots)^2
\][/tex]
we need to transform the left-hand side of the equation into a perfect square trinomial. Here’s how we can do it step-by-step:
1. Start with the expression [tex]\(x^2 + 10x\)[/tex].
2. Identify the coefficient of [tex]\(x\)[/tex], which in this case is [tex]\(10\)[/tex].
3. Take half of the coefficient of [tex]\(x\)[/tex]:
[tex]\[
\frac{10}{2} = 5
\][/tex]
4. Square that value to complete the square:
[tex]\[
5^2 = 25
\][/tex]
5. Add this squared value to the original expression to form the perfect square trinomial:
[tex]\[
x^2 + 10x + 25
\][/tex]
6. Now express this trinomial as a square of a binomial:
[tex]\[
x^2 + 10x + 25 = (x + 5)^2
\][/tex]
Thus, the completed equation is:
[tex]\[
x^2 + 10x + 25 = (x + 5)^2
\][/tex]
So, the correct answer is:
[tex]\[
D. \quad 25 ; x+5
\][/tex]
