1. For a quadratic equation ax² + bx + c = 0, the discriminant D is given by:
- Ab² - 4ac
- Bb² + 4ac
- C4ac - b²
- D2b - 4ac
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Answer: A) b² - 4ac
By definition the discriminant of ax² + bx + c = 0 is D = b² - 4ac.
Class 10 Nature of Roots (Discriminant) — practice questions with answers
Quadratic Equations · 5 sample questions from the school syllabus. Try each one, then open the answer and the worked solution.
2. The roots of 2x² + 3x - 5 = 0 are:
- AReal and distinct
- BReal and equal
- CNot real
- DEqual to zero
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Answer: A) Real and distinct
D = 9 - 4(2)(-5) = 9 + 40 = 49 > 0, so roots are real and distinct.
3. If a quadratic equation has two equal roots, the number of distinct real solutions it has is:
- A2
- B1
- C0
- D3
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Answer: B) 1
Equal roots mean both roots coincide, so there is only 1 distinct real solution.
4. Three friends in Chennai each set up a quadratic for a project:
E1: x² - 7x + 12 = 0, E2: x² - 4x + 4 = 0, E3: x² + x + 7 = 0.
How many of these have real roots (equal or distinct)?
E1: x² - 7x + 12 = 0, E2: x² - 4x + 4 = 0, E3: x² + x + 7 = 0.
How many of these have real roots (equal or distinct)?
- A2
- B3
- C4
- D1
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Answer: A) 2
E1: D = 49 - 48 = 1 > 0 (real). E2: D = 16 - 16 = 0 (real, equal). E3: D = 1 - 28 = -27 < 0 (not real). So 2 of them have real roots.
5. Prove-style reasoning: For the quadratic equation x² - 2(a+b)x + (a² + b² + c) = 0 to have equal roots, the discriminant gives a relation.
Simplify D = 0 to find what c must equal.
Simplify D = 0 to find what c must equal.
- Ac = 2ab
- Bc = a² + b²
- Cc = (a+b)²
- Dc = a*b
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Answer: A) c = 2ab
D = 0: [2(a+b)]² - 4(a²+b²+c) = 0 => 4(a+b)² = 4(a²+b²+c) => (a+b)² = a²+b²+c => a²+2ab+b² = a²+b²+c => c = 2ab.
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