https://file.aiquickdraw.com/mathbot/file/1728658364803k39bhd3d.jpg

The problem in the image shows a quadratic function, \( f(x) = 3x^2 + 21x - 36 \), and it seems to be asking to rewrite it in vertex form, which has the structure:

\[
f(x) = a(x - h)^2 + k
\]

where \( (h, k) \) is the vertex of the parabola.

To convert the given quadratic function into vertex form, we'll complete the square.

### Step 1: Factor out the coefficient of \( x^2 \) from the first two terms
\[
f(x) = 3(x^2 + 7x) - 36
\]

### Step 2: Complete the square
To complete the square inside the parentheses, we take half of the coefficient of \( x \) (which is 7), square it, and add and subtract this value inside the parentheses. Half of 7 is \( \frac{7}{2} \), and squaring it gives \( \left(\frac{7}{2}\right)^2 = \frac{49}{4} \).

\[
f(x) = 3 \left(x^2 + 7x + \frac{49}{4} - \frac{49}{4}\right) - 36
\]
\[
f(x) = 3 \left(\left(x + \frac{7}{2}\right)^2 - \frac{49}{4}\right) - 36
\]

### Step 3: Simplify the expression
Distribute the 3 and simplify the constants:
\[
f(x) = 3\left(x + \frac{7}{2}\right)^2 - 3\left(\frac{49}{4}\right) - 36
\]
\[
f(x) = 3\left(x + \frac{7}{2}\right)^2 - \frac{147}{4} - 36
\]
Convert 36 to a fraction with the denominator 4:
\[
f(x) = 3\left(x + \frac{7}{2}\right)^2 - \frac{147}{4} - \frac{144}{4}
\]
\[
f(x) = 3\left(x + \frac{7}{2}\right)^2 - \frac{291}{4}
\]

Thus, the vertex form of the function is:
\[
f(x) = 3\left(x + \frac{7}{2}\right)^2 - \frac{291}{4}
\]

Let me know if you want further details or have any questions!

### Questions:
1. How do you identify the vertex from the vertex form of a quadratic function?
2. Can you apply the method of completing the square to another quadratic equation?
3. What does the value of \( a \) (in this case 3) tell you about the parabola's shape?
4. How can the vertex form of a quadratic function be useful in solving real-world problems?
5. What is the geometric interpretation of the constant term in the vertex form?

### Tip:
When completing the square, always make sure to factor out any coefficient in front of \( x^2 \) to avoid errors in the process.

f(x) = 3x^2 + 21x - 36. Convert the given quadratic equation to its vertex form.

Learn how to convert the quadratic equation f(x) = 3x^2 + 21x - 36 to its vertex form using the method of completing the square. Detailed explanation and steps provided. Suitable for students in Grades 9-11.

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