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You are looking to convert the quadratic function \( f(x) = 3x^2 + 21x - 36 \) into the form \( f(x) = a(x - h)^2 + k \), which is called the **vertex form**. To do this, we need to complete the square. Let me walk you through the steps:

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

### Step 2: Complete the square inside the parentheses.
To complete the square, take half of the coefficient of \(x\) (which is 7), square it, and add and subtract it inside the parentheses:
\[
f(x) = 3 \left( x^2 + 7x + \left( \frac{7}{2} \right)^2 - \left( \frac{7}{2} \right)^2 \right) - 36
\]
\[
f(x) = 3 \left( x^2 + 7x + \frac{49}{4} - \frac{49}{4} \right) - 36
\]
\[
f(x) = 3 \left( \left( x + \frac{7}{2} \right)^2 - \frac{49}{4} \right) - 36
\]

### Step 3: Simplify the expression.
Now, distribute the factor of 3:
\[
f(x) = 3 \left( x + \frac{7}{2} \right)^2 - 3 \times \frac{49}{4} - 36
\]
\[
f(x) = 3 \left( x + \frac{7}{2} \right)^2 - \frac{147}{4} - 36
\]

Convert -36 to have a denominator of 4:
\[
f(x) = 3 \left( x + \frac{7}{2} \right)^2 - \frac{147}{4} - \frac{144}{4}
\]
\[
f(x) = 3 \left( x + \frac{7}{2} \right)^2 - \frac{291}{4}
\]

### Final Answer:
The quadratic function in vertex form is:
\[
f(x) = 3 \left( x + \frac{7}{2} \right)^2 - \frac{291}{4}
\]

This is the desired form \( f(x) = a(x - h)^2 + k \), where:
- \( a = 3 \),
- \( h = -\frac{7}{2} \), and
- \( k = -\frac{291}{4} \).

Would you like more details or have any questions?

Here are five related questions you could explore:
1. How do you derive the vertex from the vertex form of a quadratic equation?
2. What is the importance of completing the square in solving quadratic equations?
3. How does the coefficient \( a \) affect the shape of the parabola?
4. What is the axis of symmetry for the given function?
5. How would you find the x-intercepts of this quadratic function?

**Tip:** When completing the square, always ensure that you maintain balance in the equation by adjusting constants both inside and outside the parentheses correctly.

f(x) = 3x^2 + 21x - 36. Turn into f(x) = a(x - h)^2 + k.

Learn how to convert the quadratic function f(x) = 3x^2 + 21x - 36 into vertex form using the method of completing the square. This process involves factoring, adding and subtracting values to complete the square, and simplifying. Suitable for students in Grades 9-11.

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