Let's go through the steps to solve these problems.

Part 1: Solve the Quadratic Equations by Completing the Square

1. Equation: $n^2 - 6n = 72$

   • Move 72 to the left side: $n^2 - 6n - 72 = 0$
   • Complete the square: Rewrite in the form $(n - p)^2 = q$.
   • Solve for $n$ using square roots.

2. Equation: $w^2 - 5w - \frac{11}{4} = 0$

   • Move $\frac{11}{4}$ to the left side: $w^2 - 5w = \frac{11}{4}$
   • Complete the square: Rewrite in the form $(w - p)^2 = q$.
   • Solve for $w$ using square roots.

3. Equation: $4x^2 + 16x = 9$

   • Divide all terms by 4 to simplify: $x^2 + 4x = \frac{9}{4}$
   • Complete the square: Rewrite in the form $(x + p)^2 = q$.
   • Solve for $x$ using square roots.

4. Equation: $x^2 + 12x + 10 = 0$

   • Move 10 to the left side: $x^2 + 12x = -10$
   • Complete the square: Rewrite in the form $(x + p)^2 = q$.
   • Solve for $x$ using square roots.

5. Equation: $2x^2 - 8x - 16 = 0$

   • Divide all terms by 2 to simplify: $x^2 - 4x = 8$
   • Complete the square: Rewrite in the form $(x - p)^2 = q$.
   • Solve for $x$ using square roots.

6. Equation: $3a^2 - 12a + 3 = -a^2 - 4$

   • Move all terms to one side to form a quadratic equation in $a$: $4a^2 - 12a + 7 = 0$
   • Complete the square: Rewrite in the form $(a - p)^2 = q$.
   • Solve for $a$ using square roots.

Part 2: Write the Function in Vertex Form

7. Function: $f(x) = 2x^2 - 12x + 20$

   • Factor out the coefficient of $x^2$ from the first two terms: $f(x) = 2(x^2 - 6x) + 20$
   • Complete the square within the parentheses.
   • Write in vertex form $f(x) = a(x - h)^2 + k$.

8. Function: $f(x) = -4x^2 - 40x + 24$

   • Factor out $-4$ from the first two terms: $f(x) = -4(x^2 + 10x) + 24$
   • Complete the square within the parentheses.
   • Write in vertex form $f(x) = a(x - h)^2 + k$.

Would you like detailed step-by-step solutions for each of these problems?

Further Questions to Explore

1. How does completing the square help in converting a quadratic function to vertex form?
2. Why is it important to divide out the coefficient of $x^2$ when completing the square?
3. What are the main differences between solving a quadratic equation and converting it to vertex form?
4. How can you verify your solutions once you’ve completed the square?
5. What are the applications of vertex form in real-world problems?

Tip:

Always check your work by expanding the vertex form back to standard form to ensure accuracy!