Quadratic Equations Practice Questions
SAT · SAT Math · 149 free MCQs with instant results and detailed explanations.
149
Total
41
Easy
86
Medium
22
Hard
Start Practicing Quadratic Equations
Take a timed quiz or customize your practice session
Sample Questions from Quadratic Equations
Here are 10 sample questions. Start a quiz to get randomized questions with scoring.
Q1
Easy
What is the vertex of the quadratic equation y = 2x^2 - 8x + 5?
A.
(2, -3)
B.
(2, -7)
C.
(4, 1)
D.
(0, 5)
Show Answer & Explanation
Correct Answer: A
To find the vertex of the parabola given by the quadratic equation, we use the vertex formula x = -b/2a. Here, a = 2 and b = -8, so x = -(-8)/(2*2) = 2. Substitute x = 2 into the equation to find y: y = 2(2)^2 - 8(2) + 5 = -3. Therefore, the vertex is (2, -3).
Q2
Easy
What is the value of x in the equation x^2 - 6x + 9 = 0?
A.
3
B.
6
C.
0
D.
9
Show Answer & Explanation
Correct Answer: A
The equation x^2 - 6x + 9 can be factored as (x - 3)(x - 3) = 0, which simplifies to (x - 3)^2 = 0. Thus, x = 3 is the only solution (a double root).
Q3
Easy
What is the vertex of the quadratic function f(x) = 2xĀ² - 8x + 5?
A.
(2, -3)
B.
(2, 5)
C.
(4, 1)
D.
(4, -3)
Show Answer & Explanation
Correct Answer: A
To find the vertex, use the formula x = -b/(2a). Here, a = 2 and b = -8, so x = -(-8)/(2*2) = 2. Substitute x = 2 in f(x) to find y: f(2) = 2(2)Ā² - 8(2) + 5 = -3. Thus, the vertex is (2, -3).
Q4
Medium
If the roots of the quadratic equation xĀ² - 5x + k = 0 are both positive, what is the minimum value of k?
A.
6.25
B.
5
C.
4
D.
10
Show Answer & Explanation
Correct Answer: A
To ensure both roots are positive, the discriminant must be non-negative, and the sum of the roots must also be positive. The sum of the roots (5) is positive for any k, but to keep the product of the roots (k) positive, k must be at least 6.25.
Q5
Medium
What is the vertex of the quadratic function f(x) = 2xĀ² - 8x + 3?
A.
(2, -5)
B.
(4, -5)
C.
(2, 3)
D.
(4, 3)
Show Answer & Explanation
Correct Answer: A
To find the vertex, use the formula x = -b/(2a). Here, a = 2 and b = -8, thus x = 4. Substitute x into the equation to find f(4) = 2(4)Ā² - 8(4) + 3 = -5. Therefore, the vertex is (4, -5).
Q6
Medium
Which of the following represents the factored form of the quadratic equation xĀ² - 7x + 10?
A.
(x - 5)(x - 2)
B.
(x - 10)(x + 1)
C.
(x - 2)(x - 5)
D.
(x + 2)(x + 5)
Show Answer & Explanation
Correct Answer: C
To factor the quadratic xĀ² - 7x + 10, look for two numbers that multiply to 10 (the constant term) and add up to -7 (the coefficient of x). The numbers -2 and -5 satisfy this condition, thus the factorization is (x - 2)(x - 5).
Q7
Medium
If the equation 3xĀ² + cx + 9 = 0 has a double root, what is the value of c?
A.
6
B.
-6
C.
0
D.
-3
Show Answer & Explanation
Correct Answer: B
A double root occurs when the discriminant of the quadratic equation is zero. For the equation 3xĀ² + cx + 9 = 0, the discriminant is given by cĀ² - 4(3)(9) = 0. Solving for c gives cĀ² = 108, hence c = Ā±6. The value that maintains the form is c = -6.
Q8
Hard
If the quadratic equation x^2 - 4x + k = 0 has exactly one solution, what is the value of k?
A.
4
B.
8
C.
0
D.
16
Show Answer & Explanation
Correct Answer: A
A quadratic equation has exactly one solution when its discriminant is zero. Here, the discriminant is given by b^2 - 4ac = (-4)^2 - 4(1)(k). Setting this to zero gives 16 - 4k = 0, which simplifies to k = 4.
Q9
Hard
The roots of the quadratic equation 2x^2 + 3x + p = 0 are in the ratio 1:2. What is the value of p?
A.
3
B.
6
C.
9
D.
12
Show Answer & Explanation
Correct Answer: B
If the roots are in the ratio 1:2, let the roots be x and 2x. By Vieta's formulas, we have x + 2x = -b/a (sum of roots) and x * 2x = c/a (product of roots). Here, 3x = -3/2 and 2x^2 = p/2, leading to p = 6.
Q10
Hard
If the quadratic equation ax^2 + bx + c = 0 has roots that are both equal, which of the following must be true about the coefficients a, b, and c?
A.
b^2 - 4ac = 0
B.
b^2 - 4ac > 0
C.
a = 0
D.
c = 0
Show Answer & Explanation
Correct Answer: A
For a quadratic equation to have equal roots, the discriminant must be zero. The discriminant is given by b^2 - 4ac. Therefore, the correct condition is b^2 - 4ac = 0.
Showing 10 of 149 questions. Start a quiz to practice all questions with scoring and timer.
Practice All 149 Questions →More SAT Math Chapters
Other SAT Subjects
Quadratic Equations ā SAT SAT Math Practice Questions Online
This page contains 149 practice MCQs for the chapter Quadratic Equations in SAT SAT Math. The questions are organized by difficulty ā 41 easy, 86 medium, 22 hard ā so you can choose the right level for your preparation.
Every question includes a detailed explanation to help you understand the concept, not just memorize answers. Take a timed quiz to simulate exam conditions, or practice at your own pace with no time limit.