To determine the length of one side of a square room given its area, we start by realizing that the area of a square is equal to the length of one side squared. Given the area of the room as [tex]\(16x^2 - 24x + 9\)[/tex] square feet, we need to express this area as the square of a binomial.
Step 1: Identify the quadratic expression given:
[tex]\[ 16x^2 - 24x + 9 \][/tex]
Step 2: This expression needs to be factored into the form [tex]\((ax + b)^2\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are constants.
Step 3: Write the general form of the square of a binomial:
[tex]\[ (ax + b)^2 = a^2x^2 + 2abx + b^2 \][/tex]
Step 4: Match coefficients from the given quadratic expression [tex]\(16x^2 - 24x + 9\)[/tex] to those of [tex]\((ax + b)^2\)[/tex]:
[tex]\[ a^2x^2 + 2abx + b^2 \][/tex]
Comparing this with [tex]\(16x^2 - 24x + 9\)[/tex]:
[tex]\[ a^2 = 16 \][/tex]
[tex]\[ 2ab = -24 \][/tex]
[tex]\[ b^2 = 9 \][/tex]
Step 5: Solve for [tex]\(a\)[/tex] and [tex]\(b\)[/tex] from the matched coefficients:
[tex]\[ a^2 = 16 \][/tex]
[tex]\[ a = 4 \text{ or } a = -4 \][/tex]
Step 6: Choose [tex]\(a = 4\)[/tex] (without loss of generality, we can start with one solution and verify if it works; similar steps apply for [tex]\(a = -4\)[/tex]):
[tex]\[ b^2 = 9 \][/tex]
[tex]\[ b = 3 \text{ or } b = -3 \][/tex]
Step 7: Substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into [tex]\(2ab = -24\)[/tex] to verify:
For [tex]\(a = 4\)[/tex] and [tex]\(b = -3\)[/tex]:
[tex]\[ 2 \times 4 \times -3 = -24 \][/tex]
This is correct.
Thus, the square of the binomial [tex]\((4x - 3)\)[/tex] is:
[tex]\[ (4x - 3)^2 = 16x^2 - 24x + 9 \][/tex]
Step 8: Verify the other possible solutions:
For [tex]\(a = -4\)[/tex]:
[tex]\[ (a = -4, b = -3) \text{ gives } (-4x - 3)^2 \][/tex] which simplifies as follows:
[tex]\[ (-4x - 3)^2 = 16x^2 + 24x + 9 \][/tex] This does not match.
For [tex]\(a = 4, b = 3\)[/tex] and other combinations, they do not satisfy [tex]\(2ab = -24\)[/tex].
Therefore, the correct expression for the side of the square room that results in the given area is:
[tex]\[ (4x - 3) \][/tex]
Thus, the length of one side of the room is:
[tex]\[ (4x - 3) \text{ feet} \][/tex]
This matches only one of the provided options:
[tex]\[ (4x - 3) \text{ feet} \][/tex]
