To solve the problem, we need to find the missing terms in the factorization of the quadratic equation [tex]\(x^2 - 9x + 18\)[/tex].
First, let's look at the quadratic equation [tex]\(x^2 - 9x + 18\)[/tex] and factor it. We aim to rewrite it as a product of two binomials in the form:
[tex]\[
(x - a)(x - b)
\][/tex]
The quadratic equation expands to:
[tex]\[
x^2 - (a+b)x + ab
\][/tex]
By comparing this with the given quadratic equation [tex]\(x^2 - 9x + 18\)[/tex], we can identify:
[tex]\[
a + b = 9 \quad \text{and} \quad ab = 18
\][/tex]
We need to find two numbers, [tex]\(a\)[/tex] and [tex]\(b\)[/tex], that add up to 9 and multiply to 18.
Considering the pairs of factors of 18, we have:
[tex]\[
(1, 18), (2, 9), (3, 6), (6, 3), (9, 2), (18, 1)
\][/tex]
From these pairs, only [tex]\((3, 6)\)[/tex] and [tex]\((6, 3)\)[/tex] add up to 9. Therefore, the factorization of the quadratic equation is:
[tex]\[
x^2 - 9x + 18 = (x - 3)(x - 6)
\][/tex]
Next, we use this factorization to fill in the given table:
[tex]\[
\begin{array}{|c|c|c|}
\cline { 2 - 3 } \multicolumn{1}{c|}{} & x & -3 \\
\hline x & x^2 & ? \\
\hline -6 & ? & 18 \\
\hline
\end{array}
\][/tex]
We fill in the multiplication results for each cell:
1. The top right cell: [tex]\((-3) \times x = -3x\)[/tex]
2. The bottom left cell: [tex]\((-6) \times x = -6x\)[/tex]
The completed table should look like this:
[tex]\[
\begin{array}{|c|c|c|}
\cline { 2 - 3 } \multicolumn{1}{c|}{} & x & -3 \\
\hline x & x^2 & -3x \\
\hline -6 & -6x & 18 \\
\hline
\end{array}
\][/tex]
The missing terms are [tex]\(-3x\)[/tex] (top right cell) and [tex]\(-6x\)[/tex] (bottom left cell).
So the missing terms from the diagram are:
[tex]\[
-3x \quad \text{and} \quad -6x
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{-3x \text{ and } -6x}
\][/tex]
