To find the length of the room given the area in terms of \( x \), we start with the given area expression:
[tex]\[ 16x^2 - 24x + 9 \][/tex]
Since the room is square, this area must be the square of the length of one side of the room. Let’s denote the length of the side of the room as \( L \) and recognize that:
[tex]\[ L^2 = 16x^2 - 24x + 9 \][/tex]
We aim to find \( L \). To do this, we should express the quadratic polynomial \( 16x^2 - 24x + 9 \) as a square of a binomial. Notice that:
[tex]\[ 16x^2 - 24x + 9 \][/tex]
is actually a perfect square trinomial. To see this clearly, let's factor it:
[tex]\[ (4x - 3)^2 \][/tex]
To verify that \((4x - 3)^2\) is indeed equal to \( 16x^2 - 24x + 9 \), we expand it:
[tex]\[ (4x - 3)(4x - 3) = 4x \cdot 4x + 4x \cdot (-3) + (-3) \cdot 4x + (-3) \cdot (-3) \][/tex]
[tex]\[ = 16x^2 - 12x - 12x + 9 \][/tex]
[tex]\[ = 16x^2 - 24x + 9 \][/tex]
This matches the original expression, confirming our factoring is correct.
Hence, the length of one side of the room, \( L \), is:
[tex]\[ 4x - 3 \][/tex]
Thus, the length of the room is:
[tex]\[ \boxed{4x - 3} \text{ feet} \][/tex]
