The steps in writing f(x)=18x+3x2 in vertex form are shown, but a value is missing in the last step.

Write the function in standard form.
Factor a out of the first two terms.  f(x)=3(x2+6x)

Form a perfect square trinomial.  (six-halves) squared

f(x)=3(x2+6x+9)−3(9)

Write the trinomial as a binomial squared. f(x)=3(x+___)2−27

What is the missing value in the last step?

3
6
9
18

To solve this, let's follow the steps to convert the quadratic function \( f(x) = 18x + 3x^2 \) into vertex form:

### Step 1: Write the function in standard form
The given function is already in standard form:

\[
f(x) = 3x^2 + 18x
\]

### Step 2: Factor out the coefficient of \( x^2 \) from the first two terms
Factor out 3 from the terms involving \( x \):

\[
f(x) = 3(x^2 + 6x)
\]

### Step 3: Form a perfect square trinomial
To complete the square, take half of the coefficient of \( x \) (which is 6), square it, and add it inside the parentheses:

\[
\text{Half of } 6 \text{ is } 3. \quad 3^2 = 9
\]

So, add and subtract 9 inside the parentheses:

\[
f(x) = 3(x^2 + 6x + 9) - 3(9)
\]

### Step 4: Write the trinomial as a binomial squared
The trinomial \( x^2 + 6x + 9 \) can be factored as \( (x + 3)^2 \):

\[
f(x) = 3(x + 3)^2 - 27
\]

### Conclusion:
The missing value in the binomial is **3**. 

So, the correct answer is:

\[
\textbf{3}
\]

Do you want more details or have any questions?

Here are 5 related questions you might consider:
1. How do you find the vertex of a quadratic function in vertex form?
2. What is the significance of completing the square in solving quadratic equations?
3. How do you convert a quadratic function from vertex form back to standard form?
4. What are the steps to factor a quadratic equation?
5. How does the discriminant help in solving quadratic equations?

**Tip:** When completing the square, always remember to balance the equation by adding and subtracting the same value inside the parentheses.

Learn how to convert the quadratic function f(x) = 18x + 3x^2 to vertex form. Follow our detailed step-by-step guide, including completing the square and identifying the missing value in the final step. Suitable for high school students studying quadratic equations.

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