Quadratic Equations
Q. What is the value of k if the equation x² + kx + 16 = 0 has no real roots? (2022)
Solution
For no real roots, the discriminant must be less than zero: k² - 4*1*16 < 0, thus k < -8 or k > 8.
Correct Answer: A — -8
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Q. What is the value of k if the equation x² - 4x + k = 0 has no real roots? (2021)
Solution
For no real roots, the discriminant must be less than zero: (-4)² - 4*1*k < 0, hence k > 4.
Correct Answer: B — 6
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Q. What is the value of k if the equation x² - kx + 16 = 0 has roots 4 and 4? (2021)
Solution
Since the roots are equal, k = 4 + 4 = 8.
Correct Answer: A — 8
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Q. What is the vertex of the parabola represented by the equation y = x² - 4x + 3? (2022)
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A.
(2, -1)
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B.
(2, 1)
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C.
(1, 2)
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D.
(3, 0)
Solution
The vertex can be found using the formula x = -b/2a. Here, x = 2, and substituting back gives y = -1.
Correct Answer: A — (2, -1)
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Q. What is the vertex of the parabola represented by the equation y = x² - 6x + 8? (2023)
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A.
(3, -1)
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B.
(3, -5)
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C.
(2, -4)
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D.
(2, -2)
Solution
The vertex can be found using the formula x = -b/2a = 6/2 = 3. Substituting x = 3 into the equation gives y = -1.
Correct Answer: A — (3, -1)
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