Question
Karen was trying to factor
\[6x^2+10\]. She found that the greatest common factor of these terms was
\[2\] and made an area model:
An area model for a rectangle with a height of two and an unknown width. The rectangle is broken into two rectangles to isolate each term in the width. The first rectangle has a height of two and an area of six x squared. The second rectangle has a height of two and an area of ten.
\[6x^2\]
\[10\]
\[\text{Width}\]
\[2\]
What is the width of Karen's area model?
\[\text{Width}=\]
\[6x^2+10\]. She found that the greatest common factor of these terms was
\[2\] and made an area model:
An area model for a rectangle with a height of two and an unknown width. The rectangle is broken into two rectangles to isolate each term in the width. The first rectangle has a height of two and an area of six x squared. The second rectangle has a height of two and an area of ten.
\[6x^2\]
\[10\]
\[\text{Width}\]
\[2\]
What is the width of Karen's area model?
\[\text{Width}=\]
Answers
GPT-4o mini
To determine the width of Karen's area model, we start from the expression she was trying to factor:
\[
6x^2 + 10
\]
Since she found that the greatest common factor (GCF) of the terms is \(2\), we can factor \(2\) out from the expression:
\[
6x^2 + 10 = 2(3x^2 + 5)
\]
In the area model:
- The height is \(2\),
- The total area represented by the expression is \(6x^2 + 10\).
The area can be divided into two parts: one part has area \(6x^2\) and the other \(10\).
For the first rectangle, having an area of \(6x^2\) with a height of \(2\):
\[
\text{Area} = \text{Height} \times \text{Width}
\]
\[
6x^2 = 2 \times \text{Width}_1
\]
To find the width of this rectangle:
\[
\text{Width}_1 = \frac{6x^2}{2} = 3x^2
\]
For the second rectangle, having an area of \(10\) with a height of \(2\):
\[
\text{Area} = \text{Height} \times \text{Width}
\]
\[
10 = 2 \times \text{Width}_2
\]
To find the width of this rectangle:
\[
\text{Width}_2 = \frac{10}{2} = 5
\]
Now, the total width of the entire area model (the original width we are looking for) is:
\[
\text{Width} = \text{Width}_1 + \text{Width}_2 = 3x^2 + 5
\]
Therefore, Karen's width of the area model is:
\[
\text{Width} = 3x^2 + 5
\]
\[
6x^2 + 10
\]
Since she found that the greatest common factor (GCF) of the terms is \(2\), we can factor \(2\) out from the expression:
\[
6x^2 + 10 = 2(3x^2 + 5)
\]
In the area model:
- The height is \(2\),
- The total area represented by the expression is \(6x^2 + 10\).
The area can be divided into two parts: one part has area \(6x^2\) and the other \(10\).
For the first rectangle, having an area of \(6x^2\) with a height of \(2\):
\[
\text{Area} = \text{Height} \times \text{Width}
\]
\[
6x^2 = 2 \times \text{Width}_1
\]
To find the width of this rectangle:
\[
\text{Width}_1 = \frac{6x^2}{2} = 3x^2
\]
For the second rectangle, having an area of \(10\) with a height of \(2\):
\[
\text{Area} = \text{Height} \times \text{Width}
\]
\[
10 = 2 \times \text{Width}_2
\]
To find the width of this rectangle:
\[
\text{Width}_2 = \frac{10}{2} = 5
\]
Now, the total width of the entire area model (the original width we are looking for) is:
\[
\text{Width} = \text{Width}_1 + \text{Width}_2 = 3x^2 + 5
\]
Therefore, Karen's width of the area model is:
\[
\text{Width} = 3x^2 + 5
\]