To determine the zeros (or roots) of the function [tex]\( f(x) \)[/tex], we need to find the values of [tex]\( x \)[/tex] at which [tex]\( f(x) = 0 \)[/tex].
Let’s consider each form of the function:
(A) [tex]\( f(x)=\frac{1}{2}(x-5)^2-2 \)[/tex]
In this form, the function is written in vertex form, which emphasizes the vertex of the parabola. This form does not immediately reveal the zeros without further manipulation, as it primarily shows the vertex of the function, not the roots.
(B) [tex]\( f(x)=\frac{1}{2}(x-3)(x-7) \)[/tex]
This form is the factored form of the quadratic function. In this form, the roots can be directly seen by setting each factor to zero. Specifically:
[tex]\[ (x-3)(x-7) = 0 \][/tex]
Solving these gives the roots:
[tex]\[ x - 3 = 0 \implies x = 3 \][/tex]
[tex]\[ x - 7 = 0 \implies x = 7 \][/tex]
(C) [tex]\( f(x)=\frac{1}{2} x^2-5 x+\frac{21}{2} \)[/tex]
This form is the standard form (also known as the expanded form) of the quadratic equation. To find the roots from this form, you would typically use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\( a = \frac{1}{2} \)[/tex], [tex]\( b = -5 \)[/tex], and [tex]\( c = \frac{21}{2} \)[/tex]. This method involves more steps and is less straightforward than finding the roots from the factored form.
Since the factored form (B) directly reveals the zeros of the function by simple inspection, it is the form that most quickly and easily reveals the roots of the function.
Thus, the correct choice is (B).
One of the zeros from this form is: [tex]\( x = 3 \)[/tex].
