Solving Radical Equations

Show your work for each problem.

1.
Solve:
1. a)
2. b)
3. c)
4. d)
Answer:
Square both sides: , so . Check: ✓
2.
Solve:
Answer: 49
Square both sides: . Check: ✓
3.
What is the solution to ?
1. a)
2. b)No solution
3. c)
4. d)
Answer: No solution
The principal square root is always non-negative. Since for all real , there is no solution.
4.
Solve:
Answer: 24
1. Square both sides. What is ? 25
2. Now we have . Solve for . 24
3. Check: What is ? 5
5.
Solve:
Answer: 39
Square: . Add 3: . Check: ✓
6.
Solve:
Answer: 12
1. Square both sides. What is ? 25
2. We have . Subtract 1 from both sides. 24
3. Now . Divide by 2. 12
7.
Solve:
1. a)
2. b)
3. c)
4. d)
Answer:
Isolate: . Square: . Check: ✓
8.
Solve:
Answer: 6
Square: . Add 2: . Divide by 3: . Check: ✓
9.
Solve:
Answer: 14
1. First, isolate the radical. Add 2 to both sides. What does equal? 3
2. Square both sides. What is ? 9
3. Add 5 to both sides. What is ? 14
10.
Solve:
1. a)
2. b)
3. c)
4. d)
Answer:
Divide by 2: . Square: . Check: ✓
11.
Solve: . Enter all valid solutions, or 'none' if there are none.
Answer: 9
Square: . Rearrange: . Factor: . Solutions: or . Check : and ✓. Check : but ✗. Only is valid.
12.
Solve:
Answer: 6
1. Square both sides. The left side becomes . What does expand to? x^2-6x+9
2. Set up: . Rearrange to standard form (= 0). x^2-7x+6=0
3. Factor: . What are the two numbers? 6,1
4. Check : Is ? What is ? 3
5. Check : but . Is it valid? -2
13.
Solve: . How many valid solutions are there?
1. a)None
2. b)One:
3. c)One:
4. d)Two: and
Answer: One:
Square: . Rearrange: . Factor: . Check : ✓. Check : but ✗. Only is valid.
14.
Solve:
Answer: 10
1. Before squaring, note that is required. What is the minimum value of ? 7
2. Square both sides. The right side expands to? x^2-14x+49
3. We have . Rearrange to get the quadratic. x^2-15x+50=0
4. Factor: . Which solution satisfies ? 10
5. Check : 3
15.
Solve: . Enter the valid solution.
Answer: 0
Isolate: . Square: . Simplify: , so . Solutions: or . Check : ✓. Check : ✓. Both work! But is the integer solution.
16.
A pendulum has period seconds. If the period is 2 seconds, find the length in meters (round to 1 decimal place).
Answer: 1.0
1. Substitute : . Divide both sides by . What does equal? 1/pi
2. Square both sides. What does equal? 1/pi^2
3. Multiply by 9.8. What is approximately? () 1.0