Solving Quadratic Equations by Square Roots

Show your work for each problem.

1.
How many solutions does have?
1. a)2
2. b)0
3. c)Infinitely many
4. d)1
Answer: 2
Both and , so there are exactly 2 solutions: and .
2.
Solve . Enter the positive solution.
Answer: 6
means . The positive solution is .
3.
Solve . Enter the negative solution.
Answer: -9
means . The negative solution is .
4.
Does have any real solutions?
1. a)Yes,
2. b)Yes,
3. c)Yes,
4. d)No, because squares cannot be negative
Answer: No, because squares cannot be negative
Any real number squared is non-negative (). Since , there is no real number whose square is .
5.
Solve:
Answer: 10, -10
1. What is ? 10
2. What is the positive solution? 10
3. What is the negative solution? -10
6.
Solve . Enter the positive solution.
Answer: 5
. The positive solution is .
7.
Solve:
Answer: 8, -8
1. Add 9 to both sides. What is ? 64
2. What is ? 8
3. What are the two solutions? 8, -8
8.
What are the solutions to ?
1. a)
2. b)
3. c)
4. d)
Answer:
.
9.
Solve:
Answer: 2, -2
1. Add 12 to both sides. What do you get? 3x^2 = 12
2. Divide both sides by 3. What is ? 4
3. Take the square root. What are the solutions? 2, -2
10.
Solve . Enter the larger solution.
Answer: 3
. So or . The larger solution is .
11.
Solve:
Answer: 11, -1
1. Take the square root of both sides. What do you get? x - 5 = ±6
2. If , what is ? 11
3. If , what is ? -1
12.
Solve . What are the solutions?
1. a) and
2. b) and
3. c) and
4. d) and
Answer: and
. If : . If : .
13.
A square has an area of 196 square meters. What is the side length?
Answer: 14
If , then meters. We use only the positive value since length must be positive.
14.
Solve:
Answer: 2, -8
1. Divide both sides by 4. What is ? 25
2. Take the square root. What is ? ±5
3. If , what is ? 2
4. If , what is ? -8
15.
Which equation has NO real solutions?
1. a)
2. b)
3. c)
4. d)
Answer:
. Since squares cannot be negative, this has no real solutions. The others all have real solutions.