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Ch-4: Quadratic Equations
15 Questions
Before You Practice Try each question on your own first. Use the Show Answer button only after you have selected your option, then read the explanation and revise the related concept from your chapter notes.
1. Which of the following is a quadratic equation?
A. x² + 2x + 1 = (4 - x)² + 3
B. -2x² = (5 - x)(2x - 2/5)
C. (k + 1)x² + 3/2 x = 7, where k = -1
D. x³ - x² = (x - 1)³
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Answer: D. x³ - x² = (x - 1)³
Explanation: Expanding (x - 1)³ gives x³ - 3x² + 3x - 1. So x³ - x² = x³ - 3x² + 3x - 1 ⇒ 2x² - 3x + 1 = 0, which is of degree 2.
2. The roots of the quadratic equation 2x² - x - 6 = 0 are:
A. -2, 3/2
B. 2, -3/2
C. -2, -3/2
D. 2, 3/2
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Answer: B. 2, -3/2
Explanation: 2x² - 4x + 3x - 6 = 0 ⇒ 2x(x - 2) + 3(x - 2) = 0 ⇒ (2x + 3)(x - 2) = 0. Therefore, x = 2 or x = -3/2.
3. The discriminant of the quadratic equation 2x² - 4x + 3 = 0 is:
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Answer: A. -8
Explanation: Discriminant D = b² - 4ac = (-4)² - 4(2)(3) = 16 - 24 = -8.
4. If the equation x² + 4x + k = 0 has real and equal roots, then the value of k is:
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Answer: B. 4
Explanation: For equal roots, Discriminant D = b² - 4ac = 0. So 4² - 4(1)(k) = 0 ⇒ 16 - 4k = 0 ⇒ k = 4.
5. If one root of the quadratic equation 2x² + kx - 6 = 0 is 2, then the value of k is:
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Answer: B. -1
Explanation: Substitute x = 2 in the equation: 2(2)² + k(2) - 6 = 0 ⇒ 8 + 2k - 6 = 0 ⇒ 2k + 2 = 0 ⇒ k = -1.
6. The quadratic equation 2x² - √5x + 1 = 0 has:
A. Two distinct real roots
B. Two equal real roots
C. No real roots
D. More than 2 real roots
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Answer: C. No real roots
Explanation: D = b² - 4ac = (-√5)² - 4(2)(1) = 5 - 8 = -3. Since D < 0, the equation has no real roots.
7. The sum of the roots of the quadratic equation 3x² - 9x + 5 = 0 is:
A. 3
B. -3
C. 5/3
D. -5/3
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Answer: A. 3
Explanation: Sum of roots = -b/a = -(-9)/3 = 9/3 = 3.
8. The product of the roots of the quadratic equation 2x² - 8x + 6 = 0 is:
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Answer: C. 3
Explanation: Product of roots = c/a = 6/2 = 3.
9. Values of k for which the quadratic equation 2x² - kx + k = 0 has equal roots are:
A. 0 only
B. 4 only
C. 8 only
D. 0, 8
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Answer: D. 0, 8
Explanation: D = b² - 4ac = (-k)² - 4(2)(k) = k² - 8k. For equal roots, k² - 8k = 0 ⇒ k(k - 8) = 0 ⇒ k = 0 or k = 8.
10. Which constant should be added and subtracted to solve the quadratic equation 4x² - 3x - 5 = 0 by the method of completing the square?
A. 9/16
B. 3/16
C. 9/64
D. 3/8
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Answer: C. 9/64
Explanation: Divide by 4: x² - (3/4)x - 5/4 = 0. Half of coefficient of x is 3/8. Square of 3/8 is 9/64.
11. If 1/2 is a root of the equation x² + kx - 5/4 = 0, then the value of k is:
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Answer: A. 2
Explanation: (1/2)² + k(1/2) - 5/4 = 0 ⇒ 1/4 + k/2 - 5/4 = 0 ⇒ k/2 - 1 = 0 ⇒ k = 2.
12. A natural number, when increased by 12, equals 160 times its reciprocal. Find the number.
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Answer: A. 8
Explanation: Let number be x. x + 12 = 160/x ⇒ x² + 12x - 160 = 0 ⇒ (x + 20)(x - 8) = 0. Since x is a natural number, x = 8.
13. The quadratic formula to find the roots of ax² + bx + c = 0 is given by:
A. x = (-b ± √(b² - 4ac)) / 2a
B. x = (b ± √(b² - 4ac)) / 2a
C. x = (-b ± √(b² + 4ac)) / 2a
D. x = (-b ± √(b² - 4ac)) / a
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Answer: A. x = (-b ± √(b² - 4ac)) / 2a
Explanation: The standard quadratic formula for finding roots of ax² + bx + c = 0 is x = (-b ± √(b² - 4ac)) / (2a).
14. If the roots of (a² + b²)x² - 2(ac + bd)x + (c² + d²) = 0 are equal, then:
A. ab = cd
B. ad = bc
C. a/b = c/d
D. Both B and C
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Answer: D. Both B and C
Explanation: D = 0 gives 4(ac + bd)² - 4(a² + b²)(c² + d²) = 0 ⇒ (ad - bc)² = 0 ⇒ ad = bc, which is equivalent to a/b = c/d.
15. What is the nature of roots of the equation 9x² - 6x - 2 = 0?
A. No real roots
B. 2 equal real roots
C. 2 distinct real roots
D. More than 2 real roots
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Answer: C. 2 distinct real roots
Explanation: D = b² - 4ac = (-6)² - 4(9)(-2) = 36 + 72 = 108. Since D > 0, the equation has 2 distinct real roots.
