Quadratic Equations for Competitive Exam Success

Quadratic equations

Q. If the quadratic equation x^2 + bx + 9 = 0 has roots 3 and -3, what is the value of b?

A. 0
B. 6
C. -6
D. 9

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?

A. -8
B. -4
C. 4
D. 8

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?

A. -3
B. 2
C. 3
D. 4

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?

A. 2
B. -2
C. 4
D. -4

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?

A. 6
B. 9
C. 3
D. 0

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 b if a = 1 and c = -6?

A. -1
B. 1
C. 5
D. -5

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 ax^2 + bx + c = 0 are 3 and -2, what is the value of a if b = 5 and c = -6?

A. 1
B. 2
C. 3
D. 4

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 x^2 + 6x + k = 0 are -2 and -4, what is the value of k?

A. 4
B. 8
C. 10
D. 12

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?

A. -5
B. -6
C. -7
D. -8

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?

A. 4
B. 8
C. 16
D. 0

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?

A. -6
B. 6
C. 5
D. 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?

A. 7
B. 12
C. 10
D. 8

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?

A. 1
B. 2
C. 3
D. 6

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?

A. 0
B. 1
C. 2
D. 3

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:

A. 10
B. 7
C. 5
D. 3

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?

A. 10
B. 17
C. 20
D. 30

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?

A. -8
B. -4
C. 4
D. 8

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?

A. 1
B. 5
C. -1
D. -5

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?

A. 0
B. 1
C. 2
D. 3

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?

A. 8
B. 10
C. 12
D. 14

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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Quadratic equations

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