To write the function [tex]\( f(x) = 2x^2 - 44x + 185 \)[/tex] in vertex form, we need to complete the square. Here are the step-by-step instructions:
1. Factor out the 2 from the first two terms:
We start with the standard form and factor out the coefficient of [tex]\( x^2 \)[/tex] from the quadratic and linear terms:
[tex]\[
f(x) = 2(x^2 - 22x) + 185.
\][/tex]
2. Form a perfect square trinomial inside the parentheses:
To complete the square, we need to add and subtract a specific value inside the brackets that will make the expression a perfect square trinomial. We take half of the coefficient of [tex]\( x \)[/tex] (which is [tex]\(-22\)[/tex]), divide by 2, and then square it:
[tex]\[
\left(\frac{-22}{2}\right)^2 = (-11)^2 = 121.
\][/tex]
Then we add and subtract this square inside the parentheses:
[tex]\[
f(x) = 2(x^2 - 22x + 121 - 121) + 185.
\][/tex]
3. Rewrite the trinomial:
Now, we rewrite the trinomial as a square:
[tex]\[
f(x) = 2((x - 11)^2 - 121) + 185.
\][/tex]
4. Distribute the 2:
Next, we distribute the 2 to both terms inside the parentheses:
[tex]\[
f(x) = 2(x - 11)^2 - 2 \cdot 121 + 185,
\][/tex]
which simplifies to:
[tex]\[
f(x) = 2(x - 11)^2 - 242 + 185.
\][/tex]
5. Combine the constants:
Finally, we combine the constant terms:
[tex]\[
f(x) = 2(x - 11)^2 - 57.
\][/tex]
Thus, the function written in vertex form is:
[tex]\[
f(x) = 2(x - 11)^2 - 57.
\][/tex]
So the values for the completed vertex form are as follows: [tex]\(a = 2\)[/tex], [tex]\(h = 11\)[/tex], and [tex]\(k = -57\)[/tex].
