Final answer:
The function f(x) cannot be factored into the form (3x - a)(3x - b) using integer values for a and b based on the given quadratic expression. The single root for f(x) = 32x - 57 is x = 57/32.
Step-by-step explanation:
The student's question relates to rewriting and solving a quadratic equation. The function f(x) is given by f(x) = 32x - 28(3) + 27, and we are tasked with rewriting f(x) in the form (3x - a)(3x - b) where a and b are real constants. To achieve this, we must factor the quadratic expression.
First, simplify f(x):
f(x) = 32x - 84 + 27
f(x) = 32x - 57
Notice that 32 and 57 are both multiples of 3, so we can start by factoring out a 3:
f(x) = 3(16x - 19)
To express this in the form (3x - a)(3x - b), notice that 16 and 19 are not multiples of 3 and so cannot be factored further. Therefore, we cannot express f(x) in the requested form using integer values for a and b, and we might need to reassess the function or the request in the question.
For finding the roots of f(x), we set the equation to zero and solve for x:
0 = 32x - 57
x = 57/32
The single root of the function f(x) is x = 57/32.