Answer :
To determine which model represents the factors of the expression [tex]\(4x^2 - 9\)[/tex], we need to identify and use the correct factors.
First, identify the form of the given expression. Notice that [tex]\(4x^2 - 9\)[/tex] can be written as a difference of squares:
[tex]\[ 4x^2 - 9 = (2x)^2 - 3^2 \][/tex]
The difference of squares formula is given by:
[tex]\[ a^2 - b^2 = (a + b)(a - b) \][/tex]
Here, [tex]\(a = 2x\)[/tex] and [tex]\(b = 3\)[/tex], so we can write:
[tex]\[ 4x^2 - 9 = (2x + 3)(2x - 3) \][/tex]
Now let’s see the detailed steps involved visually, and check if any of the provided models match this solution.
We need to construct a table to confirm the factors of [tex]\(4x^2 - 9\)[/tex]:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & $+2x$ & $-3$ \\ \hline$+2x$ & $+4x^2$ & $-6x$ \\ \hline$-3$ & $-6x$ & $+9$ \\ \hline \end{tabular} \][/tex]
In this model:
- The terms along the first row and first column correspond to [tex]\(+2x\)[/tex] and [tex]\(-3\)[/tex].
- When multiplying across the rows and columns, you get [tex]\(+4x^2\)[/tex], [tex]\(-6x\)[/tex], and [tex]\(9\)[/tex].
Let’s see if any of the given tables match this pattern:
1. Original options provided:
- First table:
[tex]\[ \begin{tabular}{|c|c|c|c|c|c|} \hline & $+x$ & $+x$ & -1 & - & - \\ \hline $+x \|$ & $+x^2$ & $+x^2$ & $-x$ & $-x$ & $-x$ \\ \hline $+x$ & $+x^2$ & $+x^2$ & $-x$ & $-x$ & $-x$ \\ \hline -1 & $-x$ & $-x$ & -1 & - & - \\ \hline -1 & $-x$ & $-x$ & - & - & - \\ \hline -1 & $-x$ & $-x$ & - & - & 14 \\ \hline \end{tabular} \][/tex]
- Second table:
[tex]\[ \begin{tabular}{|c|c|c|c|c|c|} \hline & $+x$ & $+x$ & + & + & + \\ \hline $+x$ & $+x^2$ & $+x^2$ & $+x$ & $+x$ & $+x$ \\ \hline $+x$ & $+x^2$ & $+x^2$ & $-x$ & $-x$ & $-x$ \\ \hline + & $+x$ & $-x$ & - & - & - \\ \hline + & $+x$ & $-x$ & $1-$ & $1-$ & $1-$ \\ \hline \end{tabular} \][/tex]
Carefully inspecting these tables, neither table accurately represents the factors we arrived at for [tex]\(4x^2 - 9\)[/tex] using the difference of squares approach.
Therefore, based on our detailed exploration and the provided factored form (2x + 3) and (2x - 3), we confirm none of the provided models match the factors of the expression [tex]\(4x^2 - 9\)[/tex]. The correct expression is represented by [tex]\((2x + 3)(2x - 3)\)[/tex].
First, identify the form of the given expression. Notice that [tex]\(4x^2 - 9\)[/tex] can be written as a difference of squares:
[tex]\[ 4x^2 - 9 = (2x)^2 - 3^2 \][/tex]
The difference of squares formula is given by:
[tex]\[ a^2 - b^2 = (a + b)(a - b) \][/tex]
Here, [tex]\(a = 2x\)[/tex] and [tex]\(b = 3\)[/tex], so we can write:
[tex]\[ 4x^2 - 9 = (2x + 3)(2x - 3) \][/tex]
Now let’s see the detailed steps involved visually, and check if any of the provided models match this solution.
We need to construct a table to confirm the factors of [tex]\(4x^2 - 9\)[/tex]:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & $+2x$ & $-3$ \\ \hline$+2x$ & $+4x^2$ & $-6x$ \\ \hline$-3$ & $-6x$ & $+9$ \\ \hline \end{tabular} \][/tex]
In this model:
- The terms along the first row and first column correspond to [tex]\(+2x\)[/tex] and [tex]\(-3\)[/tex].
- When multiplying across the rows and columns, you get [tex]\(+4x^2\)[/tex], [tex]\(-6x\)[/tex], and [tex]\(9\)[/tex].
Let’s see if any of the given tables match this pattern:
1. Original options provided:
- First table:
[tex]\[ \begin{tabular}{|c|c|c|c|c|c|} \hline & $+x$ & $+x$ & -1 & - & - \\ \hline $+x \|$ & $+x^2$ & $+x^2$ & $-x$ & $-x$ & $-x$ \\ \hline $+x$ & $+x^2$ & $+x^2$ & $-x$ & $-x$ & $-x$ \\ \hline -1 & $-x$ & $-x$ & -1 & - & - \\ \hline -1 & $-x$ & $-x$ & - & - & - \\ \hline -1 & $-x$ & $-x$ & - & - & 14 \\ \hline \end{tabular} \][/tex]
- Second table:
[tex]\[ \begin{tabular}{|c|c|c|c|c|c|} \hline & $+x$ & $+x$ & + & + & + \\ \hline $+x$ & $+x^2$ & $+x^2$ & $+x$ & $+x$ & $+x$ \\ \hline $+x$ & $+x^2$ & $+x^2$ & $-x$ & $-x$ & $-x$ \\ \hline + & $+x$ & $-x$ & - & - & - \\ \hline + & $+x$ & $-x$ & $1-$ & $1-$ & $1-$ \\ \hline \end{tabular} \][/tex]
Carefully inspecting these tables, neither table accurately represents the factors we arrived at for [tex]\(4x^2 - 9\)[/tex] using the difference of squares approach.
Therefore, based on our detailed exploration and the provided factored form (2x + 3) and (2x - 3), we confirm none of the provided models match the factors of the expression [tex]\(4x^2 - 9\)[/tex]. The correct expression is represented by [tex]\((2x + 3)(2x - 3)\)[/tex].
