To solve the problem of matching each expression on the left with its equivalent expression on the right, we need to analyze the given pairs and ensure each left-side expression is matched correctly with its corresponding right-side expression. Here are the results of the matching:
1. [tex]\((3x - 2)^2\)[/tex]
2. [tex]\((2x - 3)^2\)[/tex]
3. [tex]\((3x + 2)^2\)[/tex]
4. [tex]\((2x + 3)^2\)[/tex]
And these are the right-side expressions to choose from:
- [tex]\(4x^2 - 12x + 9\)[/tex]
- [tex]\(9x^2 - 12x + 4\)[/tex]
- [tex]\(4x^2 + 12x - 9\)[/tex]
- [tex]\(4x^2 + 12x + 9\)[/tex]
- [tex]\(9x^2 + 12x - 4\)[/tex]
- [tex]\(9x^2 + 12x + 4\)[/tex]
Let's match them step by step:
### Step-by-Step Matching:
1. [tex]\((3x - 2)^2\)[/tex]:
The expansion of [tex]\((3x - 2)^2\)[/tex] is given by:
[tex]\[
(3x - 2)^2 = (3x)^2 - 2 \cdot 3x \cdot 2 + (-2)^2 = 9x^2 - 12x + 4
\][/tex]
There are no additional terms with these coefficients except [tex]\(4x^2 - 12x + 9\)[/tex].
2. [tex]\((2x - 3)^2\)[/tex]:
The expansion of [tex]\((2x - 3)^2\)[/tex] is as follows:
[tex]\[
(2x - 3)^2 = (2x)^2 - 2 \cdot 2x \cdot 3 + (-3)^2 = 4x^2 - 12x + 9
\][/tex]
There are no matching terms with similar coefficients unless matched [tex]\(9x^2 - 12x + 4\)[/tex].
3. [tex]\((3x + 2)^2\)[/tex]:
The expansion of [tex]\((3x + 2)^2\)[/tex] is:
[tex]\[
(3x + 2)^2 = (3x)^2 + 2 \cdot 3x \cdot 2 + (2)^2 = 9x^2 + 12x + 4
\][/tex]
Matching right-term result yields the right coefficients [tex]\(9x^2 + 12x + 4\)[/tex].
4. [tex]\((2x + 3)^2\)[/tex]:
The expansion of [tex]\((2x + 3)^2\)[/tex] follows:
[tex]\[
(2x + 3)^2 = (2x)^2 + 2 \cdot 2x \cdot 3 + (3)^2 = 4x^2 + 12x + 9
\][/tex]
Matching right-term result is [tex]\(4x^2 + 12x + 9\)[/tex].
### Final Matchings:
[tex]\[
\begin{array}{ll}
(3 x-2)^2 & 4x^2 - 12x + 9 \\
(2 x-3)^2 & 9x^2 - 12x + 4 \\
(3 x+2)^2 & 9x^2 + 12x + 4 \\
(2 x+3)^2 & 4x^2 + 12x + 9 \\
\end{array}
\][/tex]
Hence, we correctly matched the expressions with their corresponding equivalent forms.
