Algebra 1 Practice Test: Quadratic Formula and Discriminant
- Quadratic Formula and Discriminant
- 10 questions
- 13 minutes
These 10 questions use the quadratic formula and the discriminant b² − 4ac. Write a, b and c with their signs before you substitute, and put negative values in parentheses.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
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Question 1
What is the discriminant of 2x2 − 3x − 5?
Correct answer: A
Use b2 − 4ac = (−3)2 − 4(2)(−5) = 9 + 40 = 49. Missing the negative sign of c gives 9 − 40 = −31.
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Question 2
How many real solutions does x2 + 4x + 7 = 0 have?
Correct answer: C
The discriminant is 16 − 4(1)(7) = −12. A negative discriminant means no real solutions, because a negative number has no real square root.
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Question 3
Solve x2 − 6x + 4 = 0.
Correct answer: B
The formula gives x = 6 ± 36 − 162 = 6 ± 252 = 3 ± 5. Not dividing both terms by 2 leads to the other choices.
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Question 4
Solve 2x2 + 5x − 3 = 0.
Correct answer: D
The discriminant is 25 + 24 = 49, so x = −5 ± 74, which is 12 or −3. Dividing by a instead of 2a gives 1 and −6.
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Question 5
Which expression correctly applies the quadratic formula to 3x2 − x − 2 = 0?
Correct answer: A
Here a = 3, b = −1, c = −2, so −b = 1, b2 − 4ac = 1 + 24 and 2a = 6. Writing −b as −1 ignores that b is already negative.
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Question 6
For which positive value of k does x2 + kx + 9 = 0 have exactly one real solution?
Correct answer: C
Exactly one solution means the discriminant is 0: k2 − 36 = 0, so k = 6 (for positive k). 36 is the value of k2, not k.
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Question 7
Solve x2 + 2x − 1 = 0.
Correct answer: B
The formula gives −2 ± 4 + 42 = −2 ± 222 = −1 ± 2. Forgetting the minus in −b gives 1 ± 2.
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Question 8
A ball's height in feet after t seconds is h = −16t2 + 32t + 48. After how many seconds does the ball hit the ground?
Correct answer: D
Set h = 0: −16(t2 − 2t − 3) = −16(t − 3)(t + 1) = 0, so t = 3 (time cannot be negative). At 1 second the ball is at its highest point, not on the ground.
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Question 9
Which equation has two different rational solutions?
Correct answer: C
Its discriminant is 1 + 24 = 25, a positive perfect square, so the solutions (3 and −2) are rational. The discriminants of the others are 5, −16 and 0.
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Question 10
The discriminant of a quadratic function is 0. What does this tell you about its graph?
Correct answer: A
A discriminant of 0 means one repeated real root, so the vertex lies on the x-axis. A positive discriminant means two x-intercepts; a negative one means none.
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