To write the quadratic function [tex]\( f(x) = 3x^2 - 24x + 10 \)[/tex] in vertex form, let's follow through the steps:
1. Factor out the coefficient of [tex]\( x^2 \)[/tex] from the first two terms:
[tex]\[
f(x) = 3(x^2 - 8x) + 10
\][/tex]
2. Complete the square inside the parentheses. To do this, we use the formula [tex]\(\left( \frac{b}{2} \right)^2\)[/tex], where [tex]\( b \)[/tex] is the coefficient of [tex]\( x \)[/tex] in the expression [tex]\( x^2 - 8x \)[/tex]. Here, [tex]\( b = -8 \)[/tex], so:
[tex]\[
\left( \frac{-8}{2} \right)^2 = (-4)^2 = 16
\][/tex]
3. Add and subtract this square inside the parentheses:
[tex]\[
f(x) = 3(x^2 - 8x + 16 - 16) + 10
\][/tex]
This can be written as:
[tex]\[
f(x) = 3((x^2 - 8x + 16) - 16) + 10
\][/tex]
4. Rewriting the perfect square trinomial as a square of a binomial:
[tex]\[
f(x) = 3((x - 4)^2 - 16) + 10
\][/tex]
5. Distribute the 3 across the terms inside the parentheses:
[tex]\[
f(x) = 3(x - 4)^2 - 3 \times 16 + 10
\][/tex]
Simplifying further:
[tex]\[
f(x) = 3(x - 4)^2 - 48 + 10
\][/tex]
6. Combine the constants:
[tex]\[
f(x) = 3(x - 4)^2 - 38
\][/tex]
Therefore, the function written in vertex form is:
[tex]\[
f(x) = 3(x - 4)^2 - 38
\][/tex]
So, the correct choice is [tex]\( \boxed{f(x) = 3(x - 4)^2 - 38} \)[/tex].
