Quick Answer
SAT discriminant and number of solutions practice questions test whether students can use D = b2 – 4ac to decide if a quadratic equation has two real solutions, one repeated real solution, or no real solutions. This guide includes 70 SAT-style questions covering discriminant basics, graphs, real-world quadratic models, parameter questions, and student-produced responses.
What Should You Know Before Practicing Discriminant Questions?
- The discriminant of ax2 + bx + c = 0 is b2 – 4ac.
- If D > 0, the quadratic has two distinct real solutions.
- If D = 0, the quadratic has one repeated real solution.
- If D < 0, the quadratic has no real solutions.
- On graphs, real solutions are the same as x-intercepts.
- Most SAT mistakes happen when students forget the negative sign in b, use the wrong a value, or solve fully when the question only asks for the number of solutions.
In This Guide – 70 SAT Discriminant and Number of Solutions Practice Questions
- What does the SAT test in discriminant questions?
- How do SAT questions test the discriminant formula?
- How do you decide whether a quadratic has 0, 1, or 2 real solutions?
- How does the SAT use parameters in discriminant questions?
- How do discriminant questions appear in graphs and word problems?
- What do student-produced response discriminant questions look like?
- What mistakes cost students points?
- How should students study this topic in 2 weeks?
- Frequently asked questions
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What Does the SAT Test in Discriminant and Number of Solutions Questions?
Discriminant questions belong to SAT Advanced Math because they involve nonlinear equations, especially quadratic equations. Students may be asked to calculate the discriminant, decide how many real solutions exist, interpret x-intercepts on a graph, or find a parameter value that creates one solution, two solutions, or no real solutions.
The best SAT strategy is to avoid unnecessary solving. If the question asks for the number of real solutions, the discriminant usually answers it faster than factoring or using the quadratic formula.
| Discriminant Value | Number of Real Solutions | Graph Meaning | SAT Trap |
|---|---|---|---|
| D > 0 | Two distinct real solutions | The graph crosses the x-axis twice | Thinking the discriminant must be a perfect square |
| D = 0 | One repeated real solution | The graph touches the x-axis once | Counting the repeated root as two solutions |
| D < 0 | No real solutions | The graph does not touch the x-axis | Forgetting the question is asking about real solutions |
SAT strategy: Before calculating, identify a, b, and c carefully. Then decide whether the question wants the discriminant value, the number of real solutions, the sign of the discriminant, or a parameter value.
How Do SAT Questions Test the Discriminant Formula?
Start with the basic discriminant rule. On the SAT, you often do not need to solve the quadratic fully. You only need to decide whether D = b2 – 4ac is positive, zero, or negative.
DISCRIMINANT BASICS
EASY
Question 1
For the equation x2 – 5x + 6 = 0, how many real solutions are there?
Which choice is correct?
A) Two distinct real solutions
B) One real solution
C) No real solutions
D) Cannot be determined from a quadratic equation
Show full solution
Correct answer: A) two distinct real solutions
Use D = b2 – 4ac. Here a = 1, b = -5, and c = 6. The discriminant is -52 – 4(1)(6) = 1. Since D is positive, the equation has two distinct real solutions.
SAT trap: The discriminant tells the number of real solutions. It does not tell whether the solutions are easy integers unless you solve further.
DISCRIMINANT BASICS
EASY
Question 2
For the equation x2 + 4x + 4 = 0, how many real solutions are there?
Which choice is correct?
A) Two distinct real solutions
B) One real solution
C) No real solutions
D) Cannot be determined from a quadratic equation
Show full solution
Correct answer: B) one real solution
Use D = b2 – 4ac. Here a = 1, b = 4, and c = 4. The discriminant is 42 – 4(1)(4) = 0. Since D is zero, the equation has one real solution.
SAT trap: The discriminant tells the number of real solutions. It does not tell whether the solutions are easy integers unless you solve further.
DISCRIMINANT BASICS
EASY
Question 3
For the equation x2 + 2x + 5 = 0, how many real solutions are there?
Which choice is correct?
A) Two distinct real solutions
B) One real solution
C) No real solutions
D) Cannot be determined from a quadratic equation
Show full solution
Correct answer: C) no real solutions
Use D = b2 – 4ac. Here a = 1, b = 2, and c = 5. The discriminant is 22 – 4(1)(5) = -16. Since D is negative, the equation has no real solutions.
SAT trap: The discriminant tells the number of real solutions. It does not tell whether the solutions are easy integers unless you solve further.
DISCRIMINANT BASICS
EASY
Question 4
For the equation 2x2 + 3x – 2 = 0, how many real solutions are there?
Which choice is correct?
A) Two distinct real solutions
B) One real solution
C) No real solutions
D) Cannot be determined from a quadratic equation
Show full solution
Correct answer: A) two distinct real solutions
Use D = b2 – 4ac. Here a = 2, b = 3, and c = -2. The discriminant is 32 – 4(2)(-2) = 25. Since D is positive, the equation has two distinct real solutions.
SAT trap: The discriminant tells the number of real solutions. It does not tell whether the solutions are easy integers unless you solve further.
DISCRIMINANT BASICS
EASY
Question 5
For the equation 3x2 – 6x + 3 = 0, how many real solutions are there?
Which choice is correct?
A) Two distinct real solutions
B) One real solution
C) No real solutions
D) Cannot be determined from a quadratic equation
Show full solution
Correct answer: B) one real solution
Use D = b2 – 4ac. Here a = 3, b = -6, and c = 3. The discriminant is -62 – 4(3)(3) = 0. Since D is zero, the equation has one real solution.
SAT trap: The discriminant tells the number of real solutions. It does not tell whether the solutions are easy integers unless you solve further.
DISCRIMINANT BASICS
MEDIUM
Question 6
For the equation 4x2 + 4x + 5 = 0, how many real solutions are there?
Which choice is correct?
A) Two distinct real solutions
B) One real solution
C) No real solutions
D) Cannot be determined from a quadratic equation
Show full solution
Correct answer: C) no real solutions
Use D = b2 – 4ac. Here a = 4, b = 4, and c = 5. The discriminant is 42 – 4(4)(5) = -64. Since D is negative, the equation has no real solutions.
SAT trap: The discriminant tells the number of real solutions. It does not tell whether the solutions are easy integers unless you solve further.
DISCRIMINANT BASICS
EASY
Question 7
For the equation x2 – 8x + 16 = 0, how many real solutions are there?
Which choice is correct?
A) Two distinct real solutions
B) One real solution
C) No real solutions
D) Cannot be determined from a quadratic equation
Show full solution
Correct answer: B) one real solution
Use D = b2 – 4ac. Here a = 1, b = -8, and c = 16. The discriminant is -82 – 4(1)(16) = 0. Since D is zero, the equation has one real solution.
SAT trap: The discriminant tells the number of real solutions. It does not tell whether the solutions are easy integers unless you solve further.
DISCRIMINANT BASICS
MEDIUM
Question 8
For the equation 5x2 – 2x – 1 = 0, how many real solutions are there?
Which choice is correct?
A) Two distinct real solutions
B) One real solution
C) No real solutions
D) Cannot be determined from a quadratic equation
Show full solution
Correct answer: A) two distinct real solutions
Use D = b2 – 4ac. Here a = 5, b = -2, and c = -1. The discriminant is -22 – 4(5)(-1) = 24. Since D is positive, the equation has two distinct real solutions.
SAT trap: The discriminant tells the number of real solutions. It does not tell whether the solutions are easy integers unless you solve further.
DISCRIMINANT BASICS
MEDIUM
Question 9
For the equation 2x2 + 4x + 7 = 0, how many real solutions are there?
Which choice is correct?
A) Two distinct real solutions
B) One real solution
C) No real solutions
D) Cannot be determined from a quadratic equation
Show full solution
Correct answer: C) no real solutions
Use D = b2 – 4ac. Here a = 2, b = 4, and c = 7. The discriminant is 42 – 4(2)(7) = -40. Since D is negative, the equation has no real solutions.
SAT trap: The discriminant tells the number of real solutions. It does not tell whether the solutions are easy integers unless you solve further.
DISCRIMINANT BASICS
MEDIUM
Question 10
For the equation -x2 + 6x – 9 = 0, how many real solutions are there?
Which choice is correct?
A) Two distinct real solutions
B) One real solution
C) No real solutions
D) Cannot be determined from a quadratic equation
Show full solution
Correct answer: B) one real solution
Use D = b2 – 4ac. Here a = -1, b = 6, and c = -9. The discriminant is 62 – 4(-1)(-9) = 0. Since D is zero, the equation has one real solution.
SAT trap: The discriminant tells the number of real solutions. It does not tell whether the solutions are easy integers unless you solve further.
DISCRIMINANT BASICS
MEDIUM
Question 11
For the equation 3x2 + 12x + 11 = 0, how many real solutions are there?
Which choice is correct?
A) Two distinct real solutions
B) One real solution
C) No real solutions
D) Cannot be determined from a quadratic equation
Show full solution
Correct answer: A) two distinct real solutions
Use D = b2 – 4ac. Here a = 3, b = 12, and c = 11. The discriminant is 122 – 4(3)(11) = 12. Since D is positive, the equation has two distinct real solutions.
SAT trap: The discriminant tells the number of real solutions. It does not tell whether the solutions are easy integers unless you solve further.
DISCRIMINANT BASICS
MEDIUM
Question 12
For the equation 2x2 – 12x + 18 = 0, how many real solutions are there?
Which choice is correct?
A) Two distinct real solutions
B) One real solution
C) No real solutions
D) Cannot be determined from a quadratic equation
Show full solution
Correct answer: B) one real solution
Use D = b2 – 4ac. Here a = 2, b = -12, and c = 18. The discriminant is -122 – 4(2)(18) = 0. Since D is zero, the equation has one real solution.
SAT trap: The discriminant tells the number of real solutions. It does not tell whether the solutions are easy integers unless you solve further.
DISCRIMINANT BASICS
HARD
Question 13
For the equation 7x2 + x + 3 = 0, how many real solutions are there?
Which choice is correct?
A) Two distinct real solutions
B) One real solution
C) No real solutions
D) Cannot be determined from a quadratic equation
Show full solution
Correct answer: C) no real solutions
Use D = b2 – 4ac. Here a = 7, b = 1, and c = 3. The discriminant is 12 – 4(7)(3) = -83. Since D is negative, the equation has no real solutions.
SAT trap: The discriminant tells the number of real solutions. It does not tell whether the solutions are easy integers unless you solve further.
DISCRIMINANT BASICS
EASY
Question 14
For the equation x2 + 9x + 20 = 0, how many real solutions are there?
Which choice is correct?
A) Two distinct real solutions
B) One real solution
C) No real solutions
D) Cannot be determined from a quadratic equation
Show full solution
Correct answer: A) two distinct real solutions
Use D = b2 – 4ac. Here a = 1, b = 9, and c = 20. The discriminant is 92 – 4(1)(20) = 1. Since D is positive, the equation has two distinct real solutions.
SAT trap: The discriminant tells the number of real solutions. It does not tell whether the solutions are easy integers unless you solve further.
DISCRIMINANT BASICS
HARD
Question 15
For the equation 6x2 – 5x + 4 = 0, how many real solutions are there?
Which choice is correct?
A) Two distinct real solutions
B) One real solution
C) No real solutions
D) Cannot be determined from a quadratic equation
Show full solution
Correct answer: C) no real solutions
Use D = b2 – 4ac. Here a = 6, b = -5, and c = 4. The discriminant is -52 – 4(6)(4) = -71. Since D is negative, the equation has no real solutions.
SAT trap: The discriminant tells the number of real solutions. It does not tell whether the solutions are easy integers unless you solve further.
DISCRIMINANT BASICS
EASY
Question 16
For the equation x2 – 3x – 10 = 0, how many real solutions are there?
Which choice is correct?
A) Two distinct real solutions
B) One real solution
C) No real solutions
D) Cannot be determined from a quadratic equation
Show full solution
Correct answer: A) two distinct real solutions
Use D = b2 – 4ac. Here a = 1, b = -3, and c = -10. The discriminant is -32 – 4(1)(-10) = 49. Since D is positive, the equation has two distinct real solutions.
SAT trap: The discriminant tells the number of real solutions. It does not tell whether the solutions are easy integers unless you solve further.
DISCRIMINANT BASICS
MEDIUM
Question 17
For the equation 2x2 + 5x + 2 = 0, how many real solutions are there?
Which choice is correct?
A) Two distinct real solutions
B) One real solution
C) No real solutions
D) Cannot be determined from a quadratic equation
Show full solution
Correct answer: A) two distinct real solutions
Use D = b2 – 4ac. Here a = 2, b = 5, and c = 2. The discriminant is 52 – 4(2)(2) = 9. Since D is positive, the equation has two distinct real solutions.
SAT trap: The discriminant tells the number of real solutions. It does not tell whether the solutions are easy integers unless you solve further.
DISCRIMINANT BASICS
EASY
Question 18
For the equation 9x2 + 6x + 1 = 0, how many real solutions are there?
Which choice is correct?
A) Two distinct real solutions
B) One real solution
C) No real solutions
D) Cannot be determined from a quadratic equation
Show full solution
Correct answer: B) one real solution
Use D = b2 – 4ac. Here a = 9, b = 6, and c = 1. The discriminant is 62 – 4(9)(1) = 0. Since D is zero, the equation has one real solution.
SAT trap: The discriminant tells the number of real solutions. It does not tell whether the solutions are easy integers unless you solve further.
DISCRIMINANT BASICS
EASY
Question 19
For the equation x2 – x + 1 = 0, how many real solutions are there?
Which choice is correct?
A) Two distinct real solutions
B) One real solution
C) No real solutions
D) Cannot be determined from a quadratic equation
Show full solution
Correct answer: C) no real solutions
Use D = b2 – 4ac. Here a = 1, b = -1, and c = 1. The discriminant is -12 – 4(1)(1) = -3. Since D is negative, the equation has no real solutions.
SAT trap: The discriminant tells the number of real solutions. It does not tell whether the solutions are easy integers unless you solve further.
DISCRIMINANT BASICS
MEDIUM
Question 20
For the equation 4x2 – 4x + 1 = 0, how many real solutions are there?
Which choice is correct?
A) Two distinct real solutions
B) One real solution
C) No real solutions
D) Cannot be determined from a quadratic equation
Show full solution
Correct answer: B) one real solution
Use D = b2 – 4ac. Here a = 4, b = -4, and c = 1. The discriminant is -42 – 4(4)(1) = 0. Since D is zero, the equation has one real solution.
SAT trap: The discriminant tells the number of real solutions. It does not tell whether the solutions are easy integers unless you solve further.
DISCRIMINANT BASICS
EASY
Question 21
For the equation x2 – 16 = 0, how many real solutions are there?
Which choice is correct?
A) Two distinct real solutions
B) One real solution
C) No real solutions
D) Cannot be determined from a quadratic equation
Show full solution
Correct answer: A) two distinct real solutions
Use D = b2 – 4ac. Here a = 1, b = 0, and c = -16. The discriminant is 02 – 4(1)(-16) = 64. Since D is positive, the equation has two distinct real solutions.
SAT trap: The discriminant tells the number of real solutions. It does not tell whether the solutions are easy integers unless you solve further.
DISCRIMINANT BASICS
EASY
Question 22
For the equation x2 + 9 = 0, how many real solutions are there?
Which choice is correct?
A) Two distinct real solutions
B) One real solution
C) No real solutions
D) Cannot be determined from a quadratic equation
Show full solution
Correct answer: C) no real solutions
Use D = b2 – 4ac. Here a = 1, b = 0, and c = 9. The discriminant is 02 – 4(1)(9) = -36. Since D is negative, the equation has no real solutions.
SAT trap: The discriminant tells the number of real solutions. It does not tell whether the solutions are easy integers unless you solve further.
DISCRIMINANT BASICS
MEDIUM
Question 23
For the equation 3x2 – 10x + 3 = 0, how many real solutions are there?
Which choice is correct?
A) Two distinct real solutions
B) One real solution
C) No real solutions
D) Cannot be determined from a quadratic equation
Show full solution
Correct answer: A) two distinct real solutions
Use D = b2 – 4ac. Here a = 3, b = -10, and c = 3. The discriminant is -102 – 4(3)(3) = 64. Since D is positive, the equation has two distinct real solutions.
SAT trap: The discriminant tells the number of real solutions. It does not tell whether the solutions are easy integers unless you solve further.
DISCRIMINANT BASICS
HARD
Question 24
For the equation 8x2 – 6x + 5 = 0, how many real solutions are there?
Which choice is correct?
A) Two distinct real solutions
B) One real solution
C) No real solutions
D) Cannot be determined from a quadratic equation
Show full solution
Correct answer: C) no real solutions
Use D = b2 – 4ac. Here a = 8, b = -6, and c = 5. The discriminant is -62 – 4(8)(5) = -124. Since D is negative, the equation has no real solutions.
SAT trap: The discriminant tells the number of real solutions. It does not tell whether the solutions are easy integers unless you solve further.
DISCRIMINANT BASICS
EASY
Question 25
For the equation x2 + 14x + 49 = 0, how many real solutions are there?
Which choice is correct?
A) Two distinct real solutions
B) One real solution
C) No real solutions
D) Cannot be determined from a quadratic equation
Show full solution
Correct answer: B) one real solution
Use D = b2 – 4ac. Here a = 1, b = 14, and c = 49. The discriminant is 142 – 4(1)(49) = 0. Since D is zero, the equation has one real solution.
SAT trap: The discriminant tells the number of real solutions. It does not tell whether the solutions are easy integers unless you solve further.
Download More SAT Math Topic-Wise Practice Questions
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How Do You Decide Whether a Quadratic Has 0, 1, or 2 Real Solutions?
These questions connect the discriminant to graphs, repeated roots, and x-intercepts. This is where many students save time by thinking before solving.
NUMBER OF SOLUTIONS
MEDIUM
Question 26
A quadratic equation has discriminant 64. How many distinct real solutions does it have?
Which choice is correct?
A) Two
B) One
C) Zero
D) Sixty-four
Show full solution
Correct answer: A) Two
A positive discriminant means two distinct real solutions.
SAT trap: Do not enter the discriminant value when the question asks for the number of solutions.
NUMBER OF SOLUTIONS
EASY
Question 27
A quadratic equation has discriminant 0. What does this mean?
Which choice is correct?
A) Two distinct real solutions
B) One repeated real solution
C) No real solutions
D) The equation is linear
Show full solution
Correct answer: B) One repeated real solution
A discriminant of 0 means the two roots are the same, so there is one distinct real solution.
SAT trap: A repeated solution counts as one distinct answer on SAT-style questions.
NUMBER OF SOLUTIONS
EASY
Question 28
A quadratic equation has discriminant -25. Which statement is correct?
Which choice is correct?
A) It has two real solutions
B) It has one real solution
C) It has no real solutions
D) It has twenty-five real solutions
Show full solution
Correct answer: C) It has no real solutions
A negative discriminant means the graph does not cross or touch the x-axis, so there are no real solutions.
SAT trap: Complex roots may exist, but the SAT question is asking about real solutions.
NUMBER OF SOLUTIONS
MEDIUM
Question 29
A parabola touches the x-axis at exactly one point. What is true about the discriminant?
Which choice is correct?
A) D > 0
B) D = 0
C) D < 0
D) D = 1
Show full solution
Correct answer: B) D = 0
Touching the x-axis once means the equation has one repeated real solution, so D = 0.
SAT trap: Touching once is different from crossing twice.
NUMBER OF SOLUTIONS
EASY
Question 30
A parabola crosses the x-axis at two different points. What is true about D?
Which choice is correct?
A) D > 0
B) D = 0
C) D < 0
D) D cannot be used
Show full solution
Correct answer: A) D > 0
Two different x-intercepts mean two distinct real solutions, so the discriminant is positive.
SAT trap: The parabola can open upward or downward and still have a positive discriminant.
NUMBER OF SOLUTIONS
EASY
Question 31
A parabola has no x-intercepts. What is true about the discriminant?
Which choice is correct?
A) D > 0
B) D = 0
C) D < 0
D) D = 2
Show full solution
Correct answer: C) D < 0
No x-intercepts means no real solutions to f(x) = 0, so the discriminant is negative.
SAT trap: The x-intercepts of a quadratic graph are its real zeros.
NUMBER OF SOLUTIONS
EASY
Question 32
For f(x) = (x – 4)2 + 7, how many real solutions does f(x) = 0 have?
Which choice is correct?
A) Two
B) One
C) Zero
D) Seven
Show full solution
Correct answer: C) Zero
(x – 4)2 is always at least 0. Adding 7 keeps the function positive, so f(x) = 0 has no real solutions.
SAT trap: An upward-opening parabola with a positive minimum never reaches the x-axis.
NUMBER OF SOLUTIONS
EASY
Question 33
For f(x) = -(x + 1)2 + 9, how many real solutions does f(x) = 0 have?
Which choice is correct?
A) Two
B) One
C) Zero
D) Nine
Show full solution
Correct answer: A) Two
Set -(x + 1)2 + 9 = 0. Then (x + 1)2 = 9, giving two real solutions.
SAT trap: A square equal to a positive number gives both a positive and a negative root.
NUMBER OF SOLUTIONS
EASY
Question 34
Which equation has exactly one real solution?
Which choice is correct?
A) (x – 4)2 = 0
B) (x – 4)2 = 9
C) (x – 4)2 = -9
D) x2 = 16
Show full solution
Correct answer: A) (x – 4)2 = 0
A square equals 0 only when the expression inside is 0, so there is one real solution.
SAT trap: A square equal to a positive number usually gives two real values.
NUMBER OF SOLUTIONS
EASY
Question 35
Which equation has no real solution?
Which choice is correct?
A) (x + 2)2 = -5
B) (x + 2)2 = 0
C) (x + 2)2 = 5
D) x2 = 25
Show full solution
Correct answer: A) (x + 2)2 = -5
The square of a real number cannot be negative, so this equation has no real solution.
SAT trap: Any squared expression equal to a negative number is a major SAT warning sign.
How Does the SAT Use Parameters in Discriminant Questions?
Parameter questions ask for a value that makes a quadratic have two, one, or no real solutions. Translate the wording into D > 0, D = 0, or D < 0 first.
PARAMETER QUESTIONS
MEDIUM
Question 36
For what value of k does x2 + 6x + k = 0 have exactly one real solution?
Which choice is correct?
A) 6
B) 9
C) 12
D) 36
Show full solution
Correct answer: B) 9
Set D = 0: 62 – 4k = 0. So 36 – 4k = 0, and k = 9.
SAT trap: For parameter questions, translate the phrase into D > 0, D = 0, or D < 0 before solving for the parameter.
PARAMETER QUESTIONS
MEDIUM
Question 37
For what value of k does x2 – 10x + k = 0 have one real solution?
Which choice is correct?
A) 10
B) 20
C) 25
D) 100
Show full solution
Correct answer: C) 25
Set D = 0: (-10)2 – 4k = 0. So 100 – 4k = 0, and k = 25.
SAT trap: For parameter questions, translate the phrase into D > 0, D = 0, or D < 0 before solving for the parameter.
PARAMETER QUESTIONS
HARD
Question 38
For which values of k does x2 + kx + 16 = 0 have exactly one real solution?
Which choice is correct?
A) k = 4 only
B) k = 8 only
C) k = -8 only
D) k = 8 or k = -8
Show full solution
Correct answer: D) k = 8 or k = -8
Set D = 0: k2 – 64 = 0. Thus k = 8 or k = -8.
SAT trap: For parameter questions, translate the phrase into D > 0, D = 0, or D < 0 before solving for the parameter.
PARAMETER QUESTIONS
MEDIUM
Question 39
Which value of k makes 2x2 + kx + 8 = 0 have two distinct real solutions?
Which choice is correct?
A) 0
B) 4
C) 8
D) 10
Show full solution
Correct answer: D) 10
D = k2 – 64. For two distinct real solutions, D must be positive. k = 10 gives D = 36.
SAT trap: For parameter questions, translate the phrase into D > 0, D = 0, or D < 0 before solving for the parameter.
PARAMETER QUESTIONS
MEDIUM
Question 40
Which value of k makes x2 – 4x + k = 0 have no real solutions?
Which choice is correct?
A) 0
B) 3
C) 4
D) 5
Show full solution
Correct answer: D) 5
D = 16 – 4k. For no real solutions, D 4. The listed value that works is 5.
SAT trap: For parameter questions, translate the phrase into D > 0, D = 0, or D < 0 before solving for the parameter.
PARAMETER QUESTIONS
HARD
Question 41
For what value of k does kx2 + 4x + 1 = 0 have exactly one real solution, assuming k is nonzero?
Which choice is correct?
A) 1
B) 2
C) 4
D) 8
Show full solution
Correct answer: C) 4
D = 42 – 4(k)(1) = 16 – 4k. Set D = 0 to get k = 4.
SAT trap: For parameter questions, translate the phrase into D > 0, D = 0, or D < 0 before solving for the parameter.
PARAMETER QUESTIONS
HARD
Question 42
For which values of k does x2 + 2kx + 9 = 0 have one real solution?
Which choice is correct?
A) k = 3 only
B) k = -3 only
C) k = 3 or k = -3
D) k = 9 or k = -9
Show full solution
Correct answer: C) k = 3 or k = -3
D = (2k)2 – 36 = 4k2 – 36. Set D = 0 to get k2 = 9, so k = ±3.
SAT trap: For parameter questions, translate the phrase into D > 0, D = 0, or D < 0 before solving for the parameter.
PARAMETER QUESTIONS
MEDIUM
Question 43
Which value of k makes x2 + 8x + k = 0 have no real solutions?
Which choice is correct?
A) 12
B) 16
C) 20
D) -20
Show full solution
Correct answer: C) 20
D = 64 – 4k. For no real solutions, 64 – 4k 16. The listed value that works is 20.
SAT trap: For parameter questions, translate the phrase into D > 0, D = 0, or D < 0 before solving for the parameter.
PARAMETER QUESTIONS
MEDIUM
Question 44
For what value of k does 3x2 + 6x + k = 0 have exactly one real solution?
Which choice is correct?
A) 1
B) 2
C) 3
D) 6
Show full solution
Correct answer: C) 3
D = 36 – 12k. Set D = 0, so k = 3.
SAT trap: For parameter questions, translate the phrase into D > 0, D = 0, or D < 0 before solving for the parameter.
PARAMETER QUESTIONS
MEDIUM
Question 45
Which value of k makes x2 + kx + 4 = 0 have two real solutions?
Which choice is correct?
A) 0
B) 2
C) 4
D) 5
Show full solution
Correct answer: D) 5
D = k2 – 16. For k = 5, D = 9, which is positive.
SAT trap: For parameter questions, translate the phrase into D > 0, D = 0, or D < 0 before solving for the parameter.
PARAMETER QUESTIONS
MEDIUM
Question 46
For what value of k does x2 – 2x + k = 0 have one real solution?
Which choice is correct?
A) -1
B) 0
C) 1
D) 2
Show full solution
Correct answer: C) 1
D = 4 – 4k. Set D = 0 to get k = 1.
SAT trap: For parameter questions, translate the phrase into D > 0, D = 0, or D < 0 before solving for the parameter.
PARAMETER QUESTIONS
MEDIUM
Question 47
Which value of k makes 4x2 + kx + 9 = 0 have no real solutions?
Which choice is correct?
A) 10
B) 12
C) -12
D) 20
Show full solution
Correct answer: A) 10
D = k2 – 144. No real solutions require k2 < 144. The listed value 10 works.
SAT trap: For parameter questions, translate the phrase into D > 0, D = 0, or D < 0 before solving for the parameter.
PARAMETER QUESTIONS
MEDIUM
Question 48
For what value of k does kx2 – 6x + 3 = 0 have exactly one real solution?
Which choice is correct?
A) 1
B) 2
C) 3
D) 6
Show full solution
Correct answer: C) 3
D = 36 – 12k. Set D = 0, so k = 3.
SAT trap: For parameter questions, translate the phrase into D > 0, D = 0, or D < 0 before solving for the parameter.
PARAMETER QUESTIONS
MEDIUM
Question 49
If x2 + px + 25 = 0 has exactly one real solution and p is positive, what is p?
Which choice is correct?
A) 5
B) 10
C) 25
D) 50
Show full solution
Correct answer: B) 10
D = p2 – 100. Set D = 0 to get p = ±10. Since p is positive, p = 10.
SAT trap: For parameter questions, translate the phrase into D > 0, D = 0, or D < 0 before solving for the parameter.
PARAMETER QUESTIONS
MEDIUM
Question 50
For what listed value of k does x2 + (k + 2)x + 9 = 0 have exactly one real solution?
Which choice is correct?
A) 4
B) 6
C) 8
D) 10
Show full solution
Correct answer: A) 4
D = (k + 2)2 – 36. Setting D = 0 gives k + 2 = 6 or -6, so k = 4 or -8. From the choices, k = 4 works.
SAT trap: For parameter questions, translate the phrase into D > 0, D = 0, or D < 0 before solving for the parameter.
Download More SAT Math Topic-Wise Practice Questions
Build accuracy by practicing discriminant, quadratic equations, vertex form, standard form, functions, and SAT Algebra in separate topic-wise sets.
How Do Discriminant Questions Appear in Graphs and Word Problems?
Graph and word problems test the same skill in a more applied way. The discriminant tells the number of real zeros, and the context tells which solutions are meaningful.
GRAPH, WORD, AND SAT REASONING
MEDIUM
Question 51
A quadratic graph has x-intercepts at x = -1 and x = 7. What must be true about the discriminant?
Which choice is correct?
A) It is positive
B) It is zero
C) It is negative
D) It equals 7
Show full solution
Correct answer: A) It is positive
Two different x-intercepts mean two distinct real solutions, so D > 0.
SAT trap: In applied questions, first answer the algebra question. Then check what the solution means in context.
GRAPH, WORD, AND SAT REASONING
MEDIUM
Question 52
A parabola opens upward and its vertex is above the x-axis. How many real solutions does f(x) = 0 have?
Which choice is correct?
A) Two
B) One
C) Zero
D) Cannot be determined
Show full solution
Correct answer: C) Zero
The graph stays above the x-axis, so it has no real zeros.
SAT trap: In applied questions, first answer the algebra question. Then check what the solution means in context.
GRAPH, WORD, AND SAT REASONING
MEDIUM
Question 53
A parabola opens upward and its vertex is below the x-axis. How many real solutions does f(x) = 0 have?
Which choice is correct?
A) Two
B) One
C) Zero
D) Cannot be determined
Show full solution
Correct answer: A) Two
An upward-opening parabola with a vertex below the x-axis must cross the x-axis twice.
SAT trap: In applied questions, first answer the algebra question. Then check what the solution means in context.
GRAPH, WORD, AND SAT REASONING
MEDIUM
Question 54
A parabola has its vertex exactly on the x-axis. What is the discriminant?
Which choice is correct?
A) Positive
B) Zero
C) Negative
D) Cannot be determined
Show full solution
Correct answer: B) Zero
The graph touches the x-axis at one point, so it has one repeated real solution.
SAT trap: In applied questions, first answer the algebra question. Then check what the solution means in context.
GRAPH, WORD, AND SAT REASONING
MEDIUM
Question 55
A height model has D > 0 when solving h(t) = 0. What does this usually mean?
Which choice is correct?
A) The object reaches ground level at two algebraic times
B) The object never reaches ground level
C) The object touches ground level once
D) The model is not quadratic
Show full solution
Correct answer: A) The object reaches ground level at two algebraic times
D > 0 means two real solutions to the equation h(t) = 0. In a real context, check whether both times make sense.
SAT trap: In applied questions, first answer the algebra question. Then check what the solution means in context.
GRAPH, WORD, AND SAT REASONING
MEDIUM
Question 56
A profit model has D < 0 when solving P(x) = 0. What does this mean about break-even points?
Which choice is correct?
A) Two break-even points
B) One break-even point
C) No real break-even points
D) Infinitely many break-even points
Show full solution
Correct answer: C) No real break-even points
Break-even points are real zeros of P(x). A negative discriminant means none.
SAT trap: In applied questions, first answer the algebra question. Then check what the solution means in context.
GRAPH, WORD, AND SAT REASONING
MEDIUM
Question 57
A revenue equation has D = 0 when solving R(x) = 0. What is the best interpretation?
Which choice is correct?
A) The graph crosses the x-axis twice
B) The graph touches the x-axis once
C) The graph has no x-intercepts
D) The equation must be linear
Show full solution
Correct answer: B) The graph touches the x-axis once
D = 0 means one repeated real zero, which appears as a tangent point on the x-axis.
SAT trap: In applied questions, first answer the algebra question. Then check what the solution means in context.
GRAPH, WORD, AND SAT REASONING
MEDIUM
Question 58
A rectangle model gives x2 – 14x + 49 = 0. How many real values of x satisfy it?
Which choice is correct?
A) Two
B) One
C) Zero
D) Fourteen
Show full solution
Correct answer: B) One
The discriminant is 196 – 196 = 0, so the model gives one real value.
SAT trap: In applied questions, first answer the algebra question. Then check what the solution means in context.
GRAPH, WORD, AND SAT REASONING
MEDIUM
Question 59
A width equation is x2 – 10x + 16 = 0. How many real values satisfy it?
Which choice is correct?
A) Two
B) One
C) Zero
D) Ten
Show full solution
Correct answer: A) Two
The discriminant is 100 – 64 = 36, which is positive, so there are two real values.
SAT trap: In applied questions, first answer the algebra question. Then check what the solution means in context.
GRAPH, WORD, AND SAT REASONING
MEDIUM
Question 60
A ticket model gives x2 – 20x + 150 = 0. What does the discriminant show?
Which choice is correct?
A) Two real ticket values
B) One real ticket value
C) No real ticket value
D) The model is linear
Show full solution
Correct answer: C) No real ticket value
The discriminant is 400 – 600 = -200, so there are no real solutions.
SAT trap: In applied questions, first answer the algebra question. Then check what the solution means in context.
GRAPH, WORD, AND SAT REASONING
MEDIUM
Question 61
A motion equation is -2t2 + 12t – 18 = 0. How many real solutions are there?
Which choice is correct?
A) Two
B) One
C) Zero
D) Twelve
Show full solution
Correct answer: B) One
D = 122 – 4(-2)(-18) = 144 – 144 = 0, so there is one real solution.
SAT trap: In applied questions, first answer the algebra question. Then check what the solution means in context.
GRAPH, WORD, AND SAT REASONING
MEDIUM
Question 62
A distance model gives t2 – 6t + 5 = 0. How many real solutions are there?
Which choice is correct?
A) Two
B) One
C) Zero
D) Five
Show full solution
Correct answer: A) Two
D = 36 – 20 = 16, which is positive, so there are two real solutions.
SAT trap: In applied questions, first answer the algebra question. Then check what the solution means in context.
GRAPH, WORD, AND SAT REASONING
MEDIUM
Question 63
A student claims x2 – 12x + 40 = 0 has no real solutions because 40 is large. What is the best response?
Which choice is correct?
A) Correct because c is large
B) Use the discriminant; the actual reason is D = -16
C) Incorrect because every quadratic has two real solutions
D) Correct because b is negative
Show full solution
Correct answer: B) Use the discriminant; the actual reason is D = -16
D = 144 – 160 = -16, so there are no real solutions. The reason is the negative discriminant, not the size of c alone.
SAT trap: In applied questions, first answer the algebra question. Then check what the solution means in context.
GRAPH, WORD, AND SAT REASONING
MEDIUM
Question 64
Which form is fastest when the question asks whether a quadratic has zero, one, or two real solutions?
Which choice is correct?
A) Discriminant b2 – 4ac
B) Slope formula
C) Distance formula
D) Mean formula
Show full solution
Correct answer: A) Discriminant b2 – 4ac
The discriminant is designed to classify the number of real solutions without fully solving the quadratic.
SAT trap: In applied questions, first answer the algebra question. Then check what the solution means in context.
GRAPH, WORD, AND SAT REASONING
MEDIUM
Question 65
Which statement is always true if a quadratic equation has two distinct real solutions?
Which choice is correct?
A) D > 0
B) D = 0
C) D < 0
D) a = 0
Show full solution
Correct answer: A) D > 0
Two distinct real solutions occur exactly when the discriminant is positive.
SAT trap: In applied questions, first answer the algebra question. Then check what the solution means in context.
What Do Student-Produced Response Discriminant Questions Look Like?
These grid-in style questions require a number, not a multiple-choice selection. Read whether the question asks for the discriminant, the number of solutions, or a parameter value.
STUDENT-PRODUCED RESPONSE
MEDIUM
Question 66
Student-produced response: What is the discriminant of 2x2 – 7x + 3 = 0?
Which choice is correct?
Show full solution
Correct answer: 25
D = (-7)2 – 4(2)(3) = 49 – 24 = 25.
SAT trap: For grid-in questions, enter the requested value only. Do not enter extra words or units.
STUDENT-PRODUCED RESPONSE
MEDIUM
Question 67
Student-produced response: How many real solutions does x2 + 6x + 13 = 0 have?
Which choice is correct?
Show full solution
Correct answer: 0
D = 36 – 52 = -16, so the equation has 0 real solutions.
SAT trap: For grid-in questions, enter the requested value only. Do not enter extra words or units.
STUDENT-PRODUCED RESPONSE
MEDIUM
Question 68
Student-produced response: What value of k makes x2 + 12x + k = 0 have exactly one real solution?
Which choice is correct?
Show full solution
Correct answer: 36
Set D = 0: 144 – 4k = 0. So k = 36.
SAT trap: For grid-in questions, enter the requested value only. Do not enter extra words or units.
STUDENT-PRODUCED RESPONSE
MEDIUM
Question 69
Student-produced response: What is the discriminant of 4x2 + 12x + 9 = 0?
Which choice is correct?
Show full solution
Correct answer: 0
D = 144 – 144 = 0.
SAT trap: For grid-in questions, enter the requested value only. Do not enter extra words or units.
STUDENT-PRODUCED RESPONSE
HARD
Question 70
Student-produced response: What is the smallest integer k for which x2 + 4x + k = 0 has no real solutions?
Which choice is correct?
Show full solution
Correct answer: 5
D = 16 – 4k. For no real solutions, 16 – 4k 4. The smallest integer is 5.
SAT trap: For grid-in questions, enter the requested value only. Do not enter extra words or units.
What Mistakes Cost Students Points on SAT Discriminant Questions?
| Mistake | Why It Hurts | Fix |
|---|---|---|
| Dropping the sign of b | It changes the discriminant calculation | Write a, b, and c before substituting |
| Forgetting the 4ac product | Students subtract only 4 or only c | Calculate 4ac as one chunk |
| Counting a repeated root twice | D = 0 gives one distinct real solution | Remember: zero discriminant means one solution |
| Solving when classification is enough | It wastes time on the Digital SAT | Use D to decide the number of solutions first |
How Should Students Study SAT Discriminant Questions in 2 Weeks?
| Study Days | Focus | Practice Goal |
|---|---|---|
| Days 1-3 | Discriminant formula and sign rules | Do 20 quick classification questions |
| Days 4-6 | Graphs and x-intercepts | Connect D to graph behavior |
| Days 7-10 | Parameter questions | Practice D = 0, D > 0, and D < 0 setups |
| Days 11-14 | Mixed timed SAT questions | Review every mistake and write the reason |
Need a Faster SAT Math Score Improvement Plan?
Work with TestPrepKart SAT mentors to connect topic-wise practice, weekly mock tests, and mistake review into one clear score plan.
Which SAT Math Resources Should You Use Next?
| Resource | Best For | Download |
|---|---|---|
| SAT Math Topic-Wise Practice Questions | Quadratics, functions, algebra, geometry, and data analysis practice | Download Math Practice Questions |
| SAT Prep Guide E-Book | Weekly planning, SAT strategy, and concept review | Download E-Book |
| SAT Trial Class | Students who need a score plan before the next SAT date | Book Free SAT Trial |
Frequently Asked Questions About SAT Discriminant and Number of Solutions
What is the discriminant on the SAT?
The discriminant is b2 – 4ac for a quadratic equation in the form ax2 + bx + c = 0. It tells whether the equation has two, one, or no real solutions.
Does the SAT ask discriminant questions directly?
Yes. A question may ask for the discriminant value, the number of real solutions, the sign of the discriminant, or a parameter value that changes the number of solutions.
What does D > 0 mean?
D > 0 means the quadratic equation has two distinct real solutions. On a graph, the parabola crosses the x-axis twice.
What does D = 0 mean?
D = 0 means the quadratic equation has one repeated real solution. On a graph, the parabola touches the x-axis once.
What does D < 0 mean?
D < 0 means the quadratic equation has no real solutions. On a graph, the parabola does not touch or cross the x-axis.
Should I solve the quadratic if the question asks for the number of solutions?
Usually no. The discriminant is faster because it classifies the number of real solutions without requiring the actual roots.
Are discriminant questions part of SAT Advanced Math?
Yes. They connect to nonlinear equations, especially quadratic equations and real solution behavior.
What is the biggest SAT mistake in discriminant questions?
The biggest mistake is using the wrong values for a, b, and c, especially when b is negative or the equation is not yet in standard form.
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