1. Complete the following table.

[tex]\[
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$x$ & $y$ & $x^2$ & $2xy$ & $y^2$ & $x^2 + 2xy + y^2$ \\
\hline
$2a$ & $3b$ & & & & \\
\hline
$6y$ & $9z$ & & & & \\
\hline
\end{tabular}
\][/tex]



Answer :

Claro, procederemos a completar la tabla paso a paso.

Primero, tomaremos los valores de [tex]\(x\)[/tex] y [tex]\(y\)[/tex] de cada fila y realizaremos las operaciones necesarias para llenar las columnas de [tex]\(x^2\)[/tex], [tex]\(2xy\)[/tex], [tex]\(y^2\)[/tex] y [tex]\(x^2+2xy+y^2\)[/tex].

### Primera fila:

Dado [tex]\(x = 2a\)[/tex] y [tex]\(y = 3b\)[/tex]:

1. Calcular [tex]\(x^2\)[/tex]:
[tex]\[ x^2 = (2a)^2 \][/tex]

2. Calcular [tex]\(2xy\)[/tex]:
[tex]\[ 2xy = 2 \cdot (2a) \cdot (3b) \][/tex]

3. Calcular [tex]\(y^2\)[/tex]:
[tex]\[ y^2 = (3b)^2 \][/tex]

4. Calcular [tex]\(x^2 + 2xy + y^2\)[/tex]:
[tex]\[ x^2 + 2xy + y^2 = (2a)^2 + 2 \cdot (2a) \cdot (3b) + (3b)^2 \][/tex]

### Segunda fila:

Dado [tex]\(x = 6y\)[/tex] y [tex]\(y = 9z\)[/tex]:

1. Calcular [tex]\(x^2\)[/tex]:
[tex]\[ x^2 = (6y)^2 \][/tex]

2. Calcular [tex]\(2xy\)[/tex]:
[tex]\[ 2xy = 2 \cdot (6y) \cdot (9z) \][/tex]

3. Calcular [tex]\(y^2\)[/tex]:
[tex]\[ y^2 = (9z)^2 \][/tex]

4. Calcular [tex]\(x^2 + 2xy + y^2\)[/tex]:
[tex]\[ x^2 + 2xy + y^2 = (6y)^2 + 2 \cdot (6y) \cdot (9z) + (9z)^2 \][/tex]

Ahora, compilamos estos resultados para llenar la tabla.

[tex]\[ \begin{tabular}{|c|c|c|c|c|c|} \hline $x$ & $y$ & $x^2$ & $2 x y$ & $y^2$ & $x^2+2 x y+y^2$ \\ \hline $2a$ & $3b$ & ${(2a)}^2$ & $2(2a)(3b)$ & ${(3b)}^2$ & ${(2a)}^2 + 2(2a)(3b) + {(3b)}^2$ \\ \hline $6y$ & $9z$ & ${(6y)}^2$ & $2(6y)(9z)$ & ${(9z)}^2$ & ${(6y)}^2 + 2(6y)(9z) + {(9z)}^2$ \\ \hline \end{tabular} \][/tex]

Lo cual da como resultado final:
[tex]\[ \begin{tabular}{|c|c|c|c|c|c|} \hline $x$ & $y$ & $x^2$ & $2 x y$ & $y^2$ & $x^2+2 x y+y^2$ \\ \hline $2a$ & $3b$ & $(2a)^2$ & $2(2a)(3b)$ & $(3b)^2$ & $(2a)^2 + 2(2a)(3b) + (3b)^2$ \\ \hline $6y$ & $9z$ & $(6y)^2$ & $2(6y)(9z)$ & $(9z)^2$ & $(6y)^2 + 2(6y)(9z) + (9z)^2$ \\ \hline \end{tabular} \][/tex]