Q. If the quadratic equation x^2 + bx + 9 = 0 has roots 3 and -3, what is the value of b?
Solution
The sum of the roots is 3 + (-3) = 0, so b = -0.
Correct Answer: C — -6
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Q. If the quadratic equation x^2 + kx + 16 = 0 has equal roots, what is the value of k?
Solution
For equal roots, the discriminant must be zero: k^2 - 4*1*16 = 0, thus k = -8.
Correct Answer: A — -8
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Q. If the quadratic equation x^2 + mx + n = 0 has roots 1 and -3, what is the value of n?
Solution
Using Vieta's formulas, the product of the roots is n = 1 * (-3) = -3.
Correct Answer: A — -3
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Q. If the quadratic equation x^2 + mx + n = 0 has roots 1 and -3, what is the value of m?
Solution
Using Vieta's formulas, m = -(1 + (-3)) = 2.
Correct Answer: A — 2
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Q. If the quadratic equation x^2 - kx + 9 = 0 has equal roots, what is the value of k?
Solution
For equal roots, the discriminant must be zero: k^2 - 36 = 0, hence k = 6.
Correct Answer: A — 6
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Q. If the roots of the equation ax^2 + bx + c = 0 are 3 and -2, what is the value of a if b = 5 and c = -6?
Solution
Using Vieta's formulas, a = 1 since the product of the roots (3 * -2) = -6 and sum (3 + -2) = 1.
Correct Answer: A — 1
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Q. If the roots of the equation ax^2 + bx + c = 0 are 3 and -2, what is the value of b if a = 1 and c = -6?
Solution
Using the sum of roots (-b/a = 3 + (-2) = 1), we find b = -1.
Correct Answer: A — -1
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Q. If the roots of the equation x^2 + 6x + k = 0 are -2 and -4, what is the value of k?
Solution
Using the sum and product of roots: -2 + -4 = -6 and -2*-4 = k => k = 8.
Correct Answer: C — 10
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Q. If the roots of the equation x^2 + px + q = 0 are -2 and -3, what is the value of p + q?
Solution
Using Vieta's formulas, p = -(-2 - 3) = 5 and q = (-2)(-3) = 6. Therefore, p + q = 5 + 6 = 11.
Correct Answer: C — -7
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Q. If the roots of the equation x^2 + px + q = 0 are equal, what is the relationship between p and q?
-
A.
p^2 = 4q
-
B.
p^2 > 4q
-
C.
p^2 < 4q
-
D.
p + q = 0
Solution
For equal roots, the discriminant must be zero: p^2 - 4q = 0, hence p^2 = 4q.
Correct Answer: A — p^2 = 4q
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Q. If the roots of the equation x^2 - kx + 8 = 0 are equal, what is the value of k?
Solution
For equal roots, the discriminant must be zero: k^2 - 4*1*8 = 0, solving gives k = 4.
Correct Answer: A — 4
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Q. If the roots of the quadratic equation ax^2 + bx + c = 0 are 3 and -2, what is the value of c if a = 1 and b = -1?
Solution
Using the product of the roots, c = 3 * (-2) = -6.
Correct Answer: A — -6
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Q. If the roots of the quadratic equation ax^2 + bx + c = 0 are equal, what is the condition on a, b, and c?
-
A.
b^2 - 4ac > 0
-
B.
b^2 - 4ac = 0
-
C.
b^2 - 4ac < 0
-
D.
a + b + c = 0
Solution
The condition for equal roots is given by the discriminant b^2 - 4ac = 0.
Correct Answer: B — b^2 - 4ac = 0
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Q. If the roots of the quadratic equation x^2 + mx + n = 0 are 3 and 4, what is the value of m?
Solution
The sum of the roots is 3 + 4 = 7, hence m = -7.
Correct Answer: A — 7
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Q. If the roots of the quadratic equation x^2 + px + q = 0 are equal, what is the relationship between p and q?
-
A.
p^2 = 4q
-
B.
p^2 > 4q
-
C.
p^2 < 4q
-
D.
p + q = 0
Solution
For equal roots, the discriminant must be zero: p^2 - 4q = 0, hence p^2 = 4q.
Correct Answer: A — p^2 = 4q
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Q. If the roots of the quadratic equation x^2 - 3x + p = 0 are 1 and 2, what is the value of p?
Solution
Using Vieta's formulas, sum of roots = 1 + 2 = 3 and product of roots = 1*2 = 2. Thus, p = 2.
Correct Answer: D — 6
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Q. If the sum of the roots of the equation x^2 - 3x + p = 0 is 3, what is the value of p?
Solution
The sum of the roots is given by -b/a = 3. Here, -(-3)/1 = 3, so p can be any value.
Correct Answer: A — 0
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Q. The equation x^2 + 2x + 1 = 0 can be factored as:
-
A.
(x + 1)(x + 1)
-
B.
(x - 1)(x - 1)
-
C.
(x + 2)(x + 1)
-
D.
(x - 2)(x - 1)
Solution
This is a perfect square: (x + 1)^2 = 0.
Correct Answer: A — (x + 1)(x + 1)
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Q. The equation x^2 + 4x + 4 = 0 has:
-
A.
Two distinct roots
-
B.
One repeated root
-
C.
No real roots
-
D.
None of these
Solution
The discriminant is 0, indicating one repeated root.
Correct Answer: B — One repeated root
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Q. The equation x^2 - 2x + 1 = 0 has:
-
A.
Two distinct roots
-
B.
One repeated root
-
C.
No real roots
-
D.
Infinitely many roots
Solution
The discriminant is 0, indicating one repeated root.
Correct Answer: B — One repeated root
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Q. The product of the roots of the equation x^2 + 7x + 10 = 0 is:
Solution
The product of the roots is given by c/a = 10/1 = 10.
Correct Answer: A — 10
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Q. The product of the roots of the equation x^2 - 7x + k = 0 is 10. What is the value of k?
Solution
Using Vieta's formulas, the product of the roots is k = 10. Thus, k = 17.
Correct Answer: B — 17
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Q. The quadratic equation x^2 + 4x + 4 = 0 has:
-
A.
Two distinct real roots
-
B.
One real root
-
C.
No real roots
-
D.
Infinitely many roots
Solution
The discriminant is 0, indicating one real root (a repeated root).
Correct Answer: B — One real root
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Q. The quadratic equation x^2 + 6x + 9 = 0 has roots that are:
-
A.
Real and equal
-
B.
Real and distinct
-
C.
Complex
-
D.
None of these
Solution
The discriminant is 0, hence the roots are real and equal.
Correct Answer: A — Real and equal
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Q. The quadratic equation x^2 + kx + 16 = 0 has equal roots. What is the value of k?
Solution
For equal roots, the discriminant must be zero: k^2 - 4*1*16 = 0, solving gives k = -8.
Correct Answer: A — -8
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Q. The quadratic equation x^2 + px + q = 0 has roots 3 and -2. What is the value of p?
Solution
Using the sum of roots: p = -(3 + (-2)) = -1.
Correct Answer: B — 5
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Q. The quadratic equation x^2 - 3x + 2 = 0 can be factored as?
-
A.
(x-1)(x-2)
-
B.
(x-2)(x-1)
-
C.
(x+1)(x+2)
-
D.
(x-3)(x+2)
Solution
The equation factors to (x-1)(x-2) = 0.
Correct Answer: A — (x-1)(x-2)
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Q. The quadratic equation x^2 - 4x + 4 = 0 has how many distinct real roots?
Solution
The discriminant is 0, indicating one distinct real root.
Correct Answer: B — 1
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Q. The quadratic equation x^2 - 6x + 9 = 0 has how many distinct real roots?
-
A.
0
-
B.
1
-
C.
2
-
D.
Infinite
Solution
The discriminant is 0, indicating that there is exactly one distinct real root.
Correct Answer: B — 1
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Q. The quadratic equation x^2 - 6x + k = 0 has roots that differ by 2. What is the value of k?
Solution
Let the roots be r and r+2. Then, r + (r+2) = 6 and r(r+2) = k. Solving gives k = 10.
Correct Answer: B — 10
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