The Discriminant

Show your work for each problem.

1.
What is the discriminant formula for ?
1. a)
2. b)
3. c)
4. d)
Answer:
The discriminant is . It's the part under the square root in the quadratic formula .
2.
If the discriminant is positive (), how many real solutions does the equation have?
1. a)Three real solutions
2. b)Two distinct real solutions
3. c)One real solution
4. d)No real solutions
Answer: Two distinct real solutions
When , we can take the square root of a positive number, giving us two different values with , resulting in two distinct real solutions.
3.
Calculate the discriminant for .
Answer: 0
. Since , there is exactly one (repeated) solution: .
4.
Calculate the discriminant for .
Answer: 9
. Since , there are two real solutions.
5.
A quadratic equation has discriminant . What does this tell us?
1. a)No real solutions
2. b)Four real solutions
3. c)One real solution
4. d)Two real solutions
Answer: No real solutions
When , we cannot take the square root of a negative number in the real number system. The equation has no real solutions (only complex solutions with ).
6.
Calculate the discriminant for .
Answer: 49
. Since , there are two real solutions.
7.
Calculate the discriminant of and determine the nature of solutions.
Answer: -56, no real solutions
1. What is the value of ? 3
2. What is the value of ? -2
3. What is the value of ? 5
4. Calculate 4
5. Calculate 60
6. What is the discriminant? -56
8.
The parabola has its vertex above the x-axis. What can you conclude about the discriminant?
1. a)Cannot determine
2. b)
3. c)
4. d)
Answer:
Since the parabola opens upward (a > 0) and has its vertex above the x-axis, it never crosses the x-axis. This means no real roots, so . Indeed, .
9.
Find the discriminant of and interpret the result.
Answer: 0, one repeated solution
1. What is ? -1
2. What is ? 6
3. What is ? -9
4. Calculate 36
5. Calculate 36
6. What is the discriminant? 0
10.
How many real solutions does have? (Answer: 0, 1, or 2)
Answer: 0
. Since the discriminant is negative, there are no real solutions (0 real solutions).
11.
For what value of does have exactly one solution?
Answer: 9
1. For exactly one solution, what must the discriminant equal? 0
2. Write the discriminant formula with , , 36 - 4k
3. Set the discriminant equal to zero and solve: . What is ? 9
12.
For what values of does have two distinct real solutions? (Enter as inequality, e.g., m > 5 or m < -5)
Answer: m > 4 or m < -4
For two distinct solutions: . So , giving . This means or .
13.
A ball is thrown upward with height meters. Use the discriminant to determine if it reaches 50 meters.
Answer: No, discriminant is negative
1. Set and rearrange to standard form. What equation do you get? -5t^2 + 30t - 48 = 0
2. What are , , ? a=-5, b=30, c=-48
3. Calculate 900
4. Calculate 960
5. What is the discriminant? -60
6. Does the ball reach 50 meters? (yes/no) no
14.
The equation has no real solutions. Which of these could be ?
1. a)
2. b)
3. c)
4. d)
Answer:
For no real solutions: . So , giving , therefore . Only satisfies this condition.
15.
Find all values of for which has exactly one solution.
Answer: 9
1. For one solution, the discriminant must equal what? 0
2. Write the discriminant: with , , 36 - 4p
3. Set and solve for 9