Here are the questions extracted from the image:

Questions:

1. What value is needed to complete the square? $x^2 - 6x + \_\_\_\_$
2. What value is needed to complete the square? $x^2 - 2x + \_\_\_\_$
3. Solve by completing the square. Show all the work you use to arrive at the answer. $x^2 + 6x = 81$
4. Solve by completing the square. Show all the work you use to arrive at the answer. $2x^2 = 18x + 20$
5. Solve the quadratic by completing the square. Show all the work you use to arrive at the answer. $x^2 - 20x = -34$
6. Solve the quadratic by completing the square. Show all the work you use to arrive at the answer. $x^2 + 14x = -69$

Let’s tackle the questions step by step.

Question 1: Completing the square for $x^2 - 6x + \_\_\_\_$:

To complete the square:

1. Take half the coefficient of $x$, which is $-6$. Half of $-6$ is $-3$.
2. Square $-3$: $(-3)^2 = 9$.

Answer: The value needed to complete the square is 9. The expression becomes: $x^2 - 6x + 9 = (x - 3)^2$.

Question 2: Completing the square for $x^2 - 2x + \_\_\_\_$:

To complete the square:

1. Take half the coefficient of $x$, which is $-2$. Half of $-2$ is $-1$.
2. Square $-1$: $(-1)^2 = 1$.

Answer: The value needed to complete the square is 1. The expression becomes: $x^2 - 2x + 1 = (x - 1)^2$.

Would you like me to proceed step-by-step for the remaining questions or focus on any specific part?

Follow-Up Questions:

1. Why do we take half of the coefficient of $x$ to complete the square?
2. Can completing the square work for equations where the coefficient of $x^2$ is not 1?
3. How does completing the square relate to the quadratic formula?
4. Can completing the square be used for inequalities?
5. What practical applications involve completing the square?

Tip: Completing the square is a foundational skill for understanding parabolas in vertex form, making it useful in graphing and optimization problems.