Quadratic Equations
Q. For the quadratic equation 2x^2 + 4x + 2 = 0, what is the value of the discriminant? (2020)
Solution
The discriminant D = b^2 - 4ac = 4^2 - 4(2)(2) = 16 - 16 = 0.
Correct Answer: A — 0
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Q. For the quadratic equation x^2 - 4x + 4 = 0, what type of roots does it have? (2019)
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A.
Real and distinct
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B.
Real and equal
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C.
Complex
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D.
None of the above
Solution
The discriminant is 0, indicating that the roots are real and equal.
Correct Answer: B — Real and equal
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Q. For which value of k does the equation x^2 + kx + 16 = 0 have equal roots? (2019)
Solution
For equal roots, the discriminant must be zero: k^2 - 4*1*16 = 0. Solving gives k = -8.
Correct Answer: B — -4
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Q. If one root of the quadratic equation x^2 + px + q = 0 is 3, and the other root is -1, what is the value of p? (2021)
Solution
The sum of the roots is 3 + (-1) = 2, hence p = -2.
Correct Answer: A — 2
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Q. If the quadratic equation ax^2 + bx + c = 0 has roots p and q, what is the value of p + q? (2020)
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A.
-b/a
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B.
b/a
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C.
c/a
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D.
-c/a
Solution
By Vieta's formulas, the sum of the roots p + q = -b/a.
Correct Answer: A — -b/a
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Q. If the quadratic equation x^2 + 2x + 1 = 0 is solved, what are the roots? (2022)
Solution
The equation can be factored as (x + 1)(x + 1) = 0, giving the root -1 with multiplicity 2.
Correct Answer: A — -1
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Q. If the roots of the equation x^2 + 3x + k = 0 are -1 and -2, what is the value of k? (2023)
Solution
Using Vieta's formulas, k = (-1)(-2) = 2.
Correct Answer: A — 2
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Q. The quadratic equation 2x^2 - 4x + k = 0 has no real roots. What is the condition for k? (2022)
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A.
k < 0
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B.
k > 0
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C.
k > 8
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D.
k < 8
Solution
The discriminant must be less than zero: (-4)^2 - 4*2*k < 0 leads to k > 8.
Correct Answer: C — k > 8
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Q. The quadratic equation x^2 + 6x + 9 = 0 can be expressed in which of the following forms? (2020)
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A.
(x + 3)^2
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B.
(x - 3)^2
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C.
(x + 6)^2
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D.
(x - 6)^2
Solution
This is a perfect square trinomial: (x + 3)(x + 3) = 0.
Correct Answer: A — (x + 3)^2
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Q. The quadratic equation x^2 + 6x + k = 0 has roots that are both negative. What is the condition for k? (2020)
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A.
k > 9
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B.
k < 9
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C.
k = 9
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D.
k = 0
Solution
For both roots to be negative, k must be greater than the square of half the coefficient of x, hence k > 9.
Correct Answer: A — k > 9
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Q. The sum of the roots of the quadratic equation 3x^2 + 12x + 12 = 0 is equal to what? (2022)
Solution
Using Vieta's formulas, the sum of the roots is -b/a = -12/3 = -4.
Correct Answer: A — -4
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Q. What is the product of the roots of the quadratic equation x^2 - 7x + 10 = 0? (2023)
Solution
Using Vieta's formulas, the product of the roots is c/a = 10/1 = 10.
Correct Answer: A — 10
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