First, let's find the [tex]\(x\)[/tex]-intercepts of the quadratic function [tex]\(f(x) = 4x^2 - 24x + 20\)[/tex]. We do that by setting the function equal to zero and solving for [tex]\(x\)[/tex]:
[tex]\[ 4x^2 - 24x + 20 = 0 \][/tex]
Next, we apply the quadratic formula, which is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are the coefficients of the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]. Here, [tex]\(a = 4\)[/tex], [tex]\(b = -24\)[/tex], and [tex]\(c = 20\)[/tex].
1. Calculate the discriminant [tex]\(\Delta\)[/tex]:
[tex]\[ \Delta = b^2 - 4ac = (-24)^2 - 4 \cdot 4 \cdot 20 = 576 - 320 = 256 \][/tex]
2. Find the [tex]\(x\)[/tex]-intercepts using the quadratic formula:
[tex]\[ x_1 = \frac{-b + \sqrt{\Delta}}{2a} = \frac{24 + \sqrt{256}}{8} = \frac{24 + 16}{8} = \frac{40}{8} = 5 \][/tex]
[tex]\[ x_2 = \frac{-b - \sqrt{\Delta}}{2a} = \frac{24 - \sqrt{256}}{8} = \frac{24 - 16}{8} = \frac{8}{8} = 1 \][/tex]
We have the [tex]\(x\)[/tex]-intercepts [tex]\(x_1 = 5\)[/tex] and [tex]\(x_2 = 1\)[/tex].
Now, to find the average of the [tex]\(x\)[/tex]-intercepts, we use the formula:
[tex]\[ \text{Average} = \frac{x_1 + x_2}{2} \][/tex]
Substitute the values we found:
[tex]\[ \text{Average} = \frac{5 + 1}{2} = \frac{6}{2} = 3 \][/tex]
Therefore, the equation that represents the average of the [tex]\(x\)[/tex]-intercepts is:
[tex]\[\frac{1 + 5}{2} = 3\][/tex]
Thus, the correct answer is:
[tex]\[\boxed{\frac{1+5}{2}=3}\][/tex]
