Bluebook SAT Test Nonlinear Equations and Systems — Questions and Answers

Question 1: What are the solutions to x² − 5x + 6 = 0?

x = 2 and x = 3 (Correct answer)
x = −2 and x = −3
x = 1 and x = 6
x = −1 and x = −6

Correct answer: x = 2 and x = 3

Factor as (x−2)(x−3) = 0, giving x = 2 and x = 3.

Question 2: The quadratic equation x² − 4x + k = 0 has exactly one real solution. What is the value of k?

2
4 (Correct answer)
6
8

Correct answer: 4

For exactly one solution the discriminant = 0: (−4)² − 4(1)(k) = 0 → 16 = 4k → k = 4.

Question 3: Which of the following are both solutions to 2x² + 3x − 2 = 0?

x = 1/2 and x = −2 (Correct answer)
x = 2 and x = −1/2
x = 1 and x = −2
x = 1/2 and x = 2

Correct answer: x = 1/2 and x = −2

Factor: (2x−1)(x+2) = 0 → x = 1/2 or x = −2.

Question 4: How many real solutions does the equation x² + 6x + 10 = 0 have?

None (Correct answer)
Exactly one
Exactly two
Infinitely many

Correct answer: None

Discriminant = 6² − 4(1)(10) = 36 − 40 = −4 < 0, so no real solutions exist.

Question 5: What is the vertex of the parabola y = x² − 8x + 15?

(4, −1) (Correct answer)
(4, 1)
(−4, −1)
(−4, 1)

Correct answer: (4, −1)

x-vertex = −(−8)/(2·1) = 4; y = 16 − 32 + 15 = −1; vertex = (4, −1).

Question 6: Solve for x: x² = 49

x = 7 only
x = −7 only
x = 7 or x = −7 (Correct answer)
No real solution

Correct answer: x = 7 or x = −7

Taking the square root of both sides gives x = ±7.

Question 7: The parabola y = x² + bx + c passes through (0, 5) and (5, 0). What is the value of b?

−6 (Correct answer)
−5
−4
−3

Correct answer: −6

From (0,5): c = 5. From (5,0): 25 + 5b + 5 = 0 → 5b = −30 → b = −6.

Question 8: Which values of x satisfy x² − x − 12 = 0?

x = 4 and x = −3 (Correct answer)
x = −4 and x = 3
x = 6 and x = −2
x = 3 and x = 4

Correct answer: x = 4 and x = −3

Factor: (x−4)(x+3) = 0 → x = 4 or x = −3.

Question 9: For the equation 3x² − 12x + 12 = 0, what is the value of the discriminant?

0 (Correct answer)
12
36
144

Correct answer: 0

Discriminant = (−12)² − 4(3)(12) = 144 − 144 = 0.

Question 10: Using the quadratic formula, what are the solutions of x² − 6x + 7 = 0?

x = 3 ± √2 (Correct answer)
x = 6 ± √2
x = 3 ± 2√2
x = −3 ± √2

Correct answer: x = 3 ± √2

x = [6 ± √(36−28)] / 2 = [6 ± √8] / 2 = [6 ± 2√2] / 2 = 3 ± √2.

Question 11: The system y = x² and y = x + 6 is solved by substitution. What are the x-values of the intersection points?

x = 3 and x = −2 (Correct answer)
x = −3 and x = 2
x = 6 and x = −1
x = 2 and x = −6

Correct answer: x = 3 and x = −2

Set x² = x + 6 → x² − x − 6 = 0 → (x−3)(x+2) = 0 → x = 3 or x = −2.

Question 12: How many points of intersection does the system y = x² − 3 and y = 2x have?

0
1
2 (Correct answer)
3

Correct answer: 2

x² − 3 = 2x → x² − 2x − 3 = 0; discriminant = 4 + 12 = 16 > 0, so 2 intersections.

Question 13: The function f(x) = x² − 4x + 3. At which x-value does f(x) reach its minimum?

x = 1
x = 2 (Correct answer)
x = 3
x = 4

Correct answer: x = 2

The vertex x-coordinate = 4 / (2·1) = 2; this is the minimum since a > 0.

Question 14: The system y = x + 2 and y = x² − 4 has two solutions. What is the sum of their x-coordinates?

1 (Correct answer)
2
3
5

Correct answer: 1

x² − 4 = x + 2 → x² − x − 6 = 0; by Vieta's, sum of roots = −(−1)/1 = 1.

Question 15: Using the quadratic formula, what are the solutions to 2x² − 4x − 6 = 0?

x = 3 and x = −1 (Correct answer)
x = 2 and x = −3
x = 3 and x = 1
x = −3 and x = 1

Correct answer: x = 3 and x = −1

Divide by 2: x² − 2x − 3 = 0 → (x−3)(x+1) = 0 → x = 3 or x = −1.

Question 16: The parabola y = (x − 3)² − 4 intersects the x-axis at which two values of x?

x = 1 and x = 5 (Correct answer)
x = 3 and x = −3
x = 7 and x = −1
x = 5 and x = −5

Correct answer: x = 1 and x = 5

Set y = 0: (x−3)² = 4 → x−3 = ±2 → x = 5 or x = 1.

Question 17: How many solutions does the system y = x² + 3x + 2 and y = x + 2 have?

0
1
2 (Correct answer)
Infinitely many

Correct answer: 2

x² + 3x + 2 = x + 2 → x² + 2x = 0 → x(x+2) = 0 → x = 0 or x = −2; two solutions.

Question 18: The equation x² + px + 16 = 0 has two equal real roots. What is the positive value of p?

4
6
8 (Correct answer)
10

Correct answer: 8

Equal roots require discriminant = 0: p² − 64 = 0 → p = ±8; the positive value is 8.

Question 19: A parabola y = x² − 4x + c does not intersect the x-axis. Which value of c satisfies this condition?

c = 2
c = 4
c = 5 (Correct answer)
c = 1

Correct answer: c = 5

Discriminant < 0: 16 − 4c < 0 → c > 4; only c = 5 satisfies this.

Question 20: If x² − 7x + 10 = 0, what is the value of x² − 7x?

−10 (Correct answer)
7
10
17

Correct answer: −10

From the equation: x² − 7x + 10 = 0 → x² − 7x = −10.

Question 21: Which of the following systems has exactly one solution?

y = x² and y = −1
y = x² and y = 0
y = x² − 4 and y = −5
y = (x−1)² and y = 0 (Correct answer)

Correct answer: y = (x−1)² and y = 0

(x−1)² = 0 → x = 1 is the only solution; all other choices yield 0 or 2 solutions.

Question 22: Which of the following is a factor of x² + 7x + 12?

(x + 3) (Correct answer)
(x − 3)
(x + 6)
(x − 4)

Correct answer: (x + 3)

x² + 7x + 12 = (x+3)(x+4), so (x+3) is a factor.

Question 23: The line y = −3x + 5 and the parabola y = x² − 4 intersect at two points. What is the product of the x-coordinates of these two points?

−9 (Correct answer)
−3
3
9

Correct answer: −9

x² − 4 = −3x + 5 → x² + 3x − 9 = 0; by Vieta's, product of roots = −9/1 = −9.

Question 24: What is the maximum value of the function f(x) = −x² + 4x + 1?

4
5 (Correct answer)
6
7

Correct answer: 5

Vertex x = −4/(2·(−1)) = 2; f(2) = −4 + 8 + 1 = 5.

Question 25: The equation kx² − 6x + 1 = 0 has two distinct real solutions. Which of the following must be true?

k < 9 (Correct answer)
k > 9
k = 9
k ≥ 9

Correct answer: k < 9

Discriminant > 0: 36 − 4k > 0 → k < 9.

Question 26: The roots of ax² + bx + c = 0 are both negative. If a > 0, which of the following must be true?

b > 0 and c > 0 (Correct answer)
b < 0 and c > 0
b > 0 and c < 0
b < 0 and c < 0

Correct answer: b > 0 and c > 0

Sum of roots = −b/a < 0 → b > 0; product of roots = c/a > 0 → c > 0.

Question 27: What is the y-intercept of the parabola y = 2x² − 5x + 3?

y = −3
y = 0
y = 3 (Correct answer)
y = 5

Correct answer: y = 3

Set x = 0: y = 2(0)² − 5(0) + 3 = 3.

Question 28: Solve the system y = x + 2 and y = x² by substitution. What are the x-values of the solutions?

x = 2 and x = −1 (Correct answer)
x = 0 and x = 2
x = −2 and x = 1
x = 1 and x = −2

Correct answer: x = 2 and x = −1

x² = x + 2 → x² − x − 2 = 0 → (x−2)(x+1) = 0 → x = 2 or x = −1.

Bluebook SAT Test Nonlinear Equations and Systems

What are the solutions to x² − 5x + 6 = 0?

What Practice Test Geeks adds beyond an AI chat for Bluebook SAT

An AI chatbot can quiz you, but it can't reproduce the exam conditions or keep your record. Here you get:

A timed, full-length Bluebook SAT mock exam — 98 questions in 195 minutes with a 65% pass mark, modeled on the official exam's question count and time limit, followed by a score report broken down by exam domain.
A fixed Bluebook SAT question bank: the same test every attempt, so your score is comparable from one try to the next instead of changing with every regenerated quiz.
Progress saved on your device: completed tests, scores and missed questions stay in My Progress, so you can retake only what you got wrong.
A printable Bluebook SAT cheat sheet you can print or save as a PDF.
Bluebook SAT flashcards built from the same question bank.

Start the Timed Bluebook SAT Mock Exam →

These practice questions were written with AI tools. They are not real exam questions. Exam facts are checked against the official sources listed below. Not reviewed by a subject-matter expert. Editorial policy

Sources & references

Exam facts on this page (fees, format, dates, passing scores) are checked against these official sources; the date is our last check. Editorial policy

satsuite.collegeboard.org — Score release, test delivery, time limit, test dates, fees, registration, retake policy · verified Oct 2026
bluebook.collegeboard.org — Test delivery · verified Oct 2026