To fill in the blanks in the equation [tex]\((x + \ldots)^2 = x^2 + 6x + \ldots\)[/tex], we need to match the given equation with the standard form of a binomial expansion. Let's follow these steps:
1. Understand the Binomial Expansion:
The general form of the binomial expansion for [tex]\((x + a)^2\)[/tex] is:
[tex]\[
(x + a)^2 = x^2 + 2ax + a^2
\][/tex]
2. Compare with the Given Equation:
We compare this to the given equation:
[tex]\[
(x + \ldots)^2 = x^2 + 6x + \ldots
\][/tex]
3. Identify the Coefficient of the Linear Term:
From the expansion, the coefficient of the [tex]\(x\)[/tex] term in [tex]\((x + a)^2\)[/tex] is [tex]\(2a\)[/tex]. In the given equation, the coefficient of [tex]\(x\)[/tex] is 6. Therefore, we set up the equation:
[tex]\[
2a = 6
\][/tex]
4. Solve for [tex]\(a\)[/tex]:
[tex]\[
a = \frac{6}{2} = 3
\][/tex]
5. Identify the Constant Term:
The constant term in the binomial expansion [tex]\((x + a)^2\)[/tex] is [tex]\(a^2\)[/tex]. So, substituting [tex]\(a = 3\)[/tex], we get:
[tex]\[
a^2 = 3^2 = 9
\][/tex]
6. Fill in the Blanks:
Thus, the modified equation is:
[tex]\[
(x + 3)^2 = x^2 + 6x + 9
\][/tex]
Therefore, the numbers to fill in the blanks are [tex]\(\boxed{3}\)[/tex] and [tex]\(\boxed{9}\)[/tex].
