Q. If the quadratic equation ax^2 + bx + c = 0 has roots p and q, which of the following is correct?
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A.
p + q = -b/a and pq = c/a
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B.
p + q = b/a and pq = -c/a
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C.
p + q = c/a and pq = -b/a
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D.
p + q = -c/a and pq = b/a
Solution
According to Vieta's formulas, the sum of the roots p and q is -b/a and the product is c/a.
Correct Answer: A — p + q = -b/a and pq = c/a
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Q. If the quadratic equation x^2 - 4x + k = 0 has equal roots, what is the value of k?
Solution
For equal roots, the discriminant must be zero: (-4)^2 - 4*1*k = 0, leading to k = 4.
Correct Answer: B — 4
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Q. In a quadratic equation, if the discriminant is negative, what can be inferred about the roots?
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A.
The roots are real and distinct.
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B.
The roots are real and equal.
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C.
The roots are complex and conjugate.
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D.
The roots are imaginary.
Solution
A negative discriminant indicates that the roots are complex and conjugate.
Correct Answer: C — The roots are complex and conjugate.
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Q. In the context of quadratic equations, which of the following statements is true?
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A.
The roots of a quadratic equation can be both real and equal.
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B.
A quadratic equation can have more than two roots.
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C.
The graph of a quadratic equation is a straight line.
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D.
The discriminant of a quadratic equation is always positive.
Solution
The roots of a quadratic equation can be both real and equal when the discriminant is zero.
Correct Answer: A — The roots of a quadratic equation can be both real and equal.
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Q. In the quadratic equation x^2 + 6x + 9 = 0, what type of roots does it have?
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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.
Imaginary
Solution
The discriminant is zero, indicating that the roots are real and equal.
Correct Answer: B — Real and equal
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Q. What is the sum of the roots of the quadratic equation 2x^2 - 8x + 6 = 0?
Solution
The sum of the roots is given by -b/a = 8/2 = 4.
Correct Answer: B — 4
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Q. Which of the following expressions represents the vertex of the quadratic equation y = ax^2 + bx + c?
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A.
(-b/2a, f(-b/2a))
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B.
(b/2a, f(b/2a))
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C.
(c/a, 0)
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D.
(0, c)
Solution
The vertex of the quadratic equation is given by the point (-b/2a, f(-b/2a)).
Correct Answer: A — (-b/2a, f(-b/2a))
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Q. Which of the following is the correct factorization of the quadratic equation x^2 - 5x + 6?
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A.
(x - 2)(x - 3)
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B.
(x + 2)(x + 3)
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C.
(x - 1)(x - 6)
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D.
(x + 1)(x + 6)
Solution
The quadratic x^2 - 5x + 6 factors to (x - 2)(x - 3).
Correct Answer: A — (x - 2)(x - 3)
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Q. Which of the following statements about the graph of a quadratic function is true?
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A.
It is always a parabola that opens upwards.
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B.
It can be a straight line.
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C.
It can intersect the x-axis at three points.
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D.
It is symmetric about its vertex.
Solution
The graph of a quadratic function is symmetric about its vertex.
Correct Answer: D — It is symmetric about its vertex.
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