To find the correct expression representing the width of the kitchen floor in terms of [tex]\( x \)[/tex], we need to follow these steps:
1. Understand the given area expression:
The given area of the kitchen floor is [tex]\( 4x^2 + 24x + 27 \)[/tex].
2. Determine the length of the kitchen floor:
It is given that the length [tex]\( l \)[/tex] of the floor is 9 feet more than twice the length of a tile [tex]\( x \)[/tex].
Therefore, the length [tex]\( l \)[/tex] can be expressed as:
[tex]\[
l = 2x + 9
\][/tex]
3. Find the width by dividing the area by the length:
The area of a rectangle is given by the formula:
[tex]\[
\text{Area} = \text{Length} \times \text{Width}
\][/tex]
Let [tex]\( w \)[/tex] represent the width. Then,
[tex]\[
4x^2 + 24x + 27 = (2x + 9) \times w
\][/tex]
To find [tex]\( w \)[/tex], we divide the area expression by the length:
[tex]\[
w = \frac{4x^2 + 24x + 27}{2x + 9}
\][/tex]
4. Factor the quadratic expression:
We need to factor [tex]\( 4x^2 + 24x + 27 \)[/tex]. Let's try to factor it:
[tex]\[
4x^2 + 24x + 27 = (2x + 3)(2x + 9)
\][/tex]
We can check this factorization by expanding it back:
[tex]\[
(2x + 3)(2x + 9) = 2x \cdot 2x + 2x \cdot 9 + 3 \cdot 2x + 3 \cdot 9 = 4x^2 + 18x + 6x + 27 = 4x^2 + 24x + 27
\][/tex]
The factorization is correct.
5. Determine the width:
Recall, we had:
[tex]\[
4x^2 + 24x + 27 = (2x + 9) \times w
\][/tex]
Substituting the factorized form:
[tex]\[
(2x + 3)(2x + 9) = (2x + 9) \times w
\][/tex]
We can see that:
[tex]\[
w = 2x + 3
\][/tex]
Therefore, the expression representing the width [tex]\( w \)[/tex] of the kitchen in terms of [tex]\( x \)[/tex] is:
[tex]\[
2x + 3
\][/tex]
Thus, the correct answer is [tex]\( 2x + 3 \)[/tex].
