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.