Completing the Square

Show your work for each problem.

1.
What value completes the square for ?
1. a)
2. b)
3. c)
4. d)
Answer:
To complete the square: . So .
2.
What value completes the square for ?
Answer: 25
. Therefore .
3.
Which expression equals ?
1. a)
2. b)
3. c)
4. d)
Answer:
. The middle term is .
4.
Complete: . What goes in the first blank?
Answer: 36
. So .
5.
Write in completed square form.
Answer: (x+2)^2 - 4
1. What is half of 4? 2
2. What is ? 4
3. Write the perfect square part: . What is ? 2
4. What constant do we subtract? 4
6.
What is the vertex form of ?
1. a)
2. b)
3. c)
4. d)
Answer:
7.
What is the y-coordinate of the vertex of ?
Answer: -6
. The vertex is , so the y-coordinate is .
8.
Solve by completing the square.
Answer: 1, -5
1. Move the constant to the right side: 5
2. What value completes the square? 4
3. Add to both sides: 9
4. Take the square root: 3
5. What are the two solutions? Enter the larger one. 1
9.
What value completes the square for ?
Answer: 49
. So .
10.
What is the x-coordinate of the vertex of ?
1. a)
2. b)
3. c)
4. d)
Answer:
In vertex form: . The vertex is at , so x-coordinate is .
11.
Convert to vertex form.
Answer: (x-4)^2 - 4
1. What value completes the square? 16
2. Add and subtract 16: . What is the constant? -4
3. Write the perfect square: 4
12.
What is the vertex form of ?
1. a)
2. b)
3. c)
4. d)
Answer:
13.
Solve by completing the square. Give the solutions in exact form.
Answer: 3 + sqrt(5), 3 - sqrt(5)
1. Move the constant: -4
2. What value completes the square? 9
3. After adding 9 to both sides: 5
4. Take the square root: +-sqrt(5)
14.
For , what is the y-coordinate of the vertex?
Answer: -5
. Vertex y-coordinate is .
15.
Convert to vertex form.
Answer: -(x-3)^2 + 5
1. Factor out -1: . What goes in the parentheses? x^2 - 6x
2. What value completes the square inside? 9
3. After completing the square: . What is the constant? 5
16.
A ball's height is given by meters. What is the maximum height?
1. a) meters
2. b) meters
3. c) meters
4. d) meters
Answer: meters
. Maximum height is 23 meters at t = 2 seconds.