The Quadratic Formula

Show your work for each problem.

1.
In the equation , what are the values of , , and ?
1. a), ,
2. b), ,
3. c), ,
4. d), ,
Answer: , ,
In : is the coefficient of (which is 1), is the coefficient of (which is 5), and is the constant (which is 6).
2.
Calculate the discriminant for .
Answer: 4
3.
If the discriminant , how many real solutions does the equation have?
1. a)One solution
2. b)Two solutions
3. c)Three solutions
4. d)No real solutions
Answer: Two solutions
Since , the equation has two distinct real solutions. The in the formula gives us two different values.
4.
What is if ?
Answer: 3
. This is a critical step when using the quadratic formula with negative values.
5.
Solve using the quadratic formula.
Answer: x = 2 or x = 3
1. What is the value of ? 1
2. What is the value of ? -5
3. Calculate the discriminant : 1
4. What is the larger solution? 3
6.
Calculate the discriminant for .
Answer: 49
. Note: .
7.
What does it mean if the discriminant ?
1. a)The equation has no solutions
2. b)The equation cannot be solved
3. c)The equation has exactly one real solution (double root)
4. d)The equation has two different solutions
Answer: The equation has exactly one real solution (double root)
When , . Both and give the same value, so there's one repeated solution.
8.
Solve using the quadratic formula.
Answer: x = 1/2 or x = -3
1. What is ? 2
2. What is ? 5
3. What is ? -3
4. What is the discriminant? 49
5. What is the positive solution (as a decimal)? 0.5
9.
Which equation has no real solutions?
1. a)
2. b)
3. c)
4. d)
Answer:
For : . The other equations have .
10.
Solve using the quadratic formula.
Answer: x = 3
1. What is the discriminant? 0
2. What is ? 6
3. What is ? 2
4. What is the only solution? 3
11.
Solve . Enter the positive solution as a fraction (e.g., 5/3).
Answer: 5/3
or . The positive solution is .
12.
A ball is thrown upward with height meters. When does it hit the ground?
Answer: t = 6 seconds
1. Set . What equation do you get? -5t^2 + 30t = 0
2. What is ? -5
3. What is ? 30
4. What is ? 0
5. Calculate (the non-zero solution): 6
13.
What are the solutions to ?
1. a)
2. b)
3. c)
4. d)
Answer:
14.
For what value of does have exactly one solution?
Answer: 6
. The positive value is .
15.
Solve and verify by factoring.
Answer: x = 3/2
1. What is the discriminant? 0
2. Since , how many solutions are there? 1
3. Calculate : 12
4. Calculate : 8
5. What is (as a fraction)? 3/2