Nature of roots using the discriminant: Class 10 practice
For ax² + bx + c = 0, D = b² − 4ac. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Squares
- Solving simple equations
The idea behind nature of roots using the discriminant
For ax² + bx + c = 0, D = b² − 4ac. D > 0: two distinct real roots; D = 0: two equal real roots; D < 0: no real roots. D also helps find a missing coefficient for equal roots.
Step-by-step method
- Identify a, b, c.
- Compute D = b² − 4ac.
- Read the nature of roots from the sign of D.
- For 'equal roots', set D = 0 and solve.
Worked example 1 (Basic)
Find the nature of the roots of 3x² + 2x − 1 = 0.
- D = (2)² − 4(3)(-1) = 16
- Two distinct real roots
- Answer: Two distinct real roots
Worked example 2 (Standard)
Find the nature of the roots of 2x² + 5x − 4 = 0.
- D = (5)² − 4(2)(-4) = 57
- Two distinct real roots
- Answer: Two distinct real roots
Worked example 3 (Challenge)
Find k so that 1x² + kx + 25 = 0 has equal roots.
- D = k² − 4(1)(25) = 0
- k² = 100, k = ±10
- Answer: k = ±10
Common mistakes and misconceptions
- Forgetting the minus sign in −4ac
- Missing the negative value of k
Practice questions
- (Basic) Find the nature of the roots of 4x² − 4x − 1 = 0.
- (Basic) Find the nature of the roots of 1x² − 4x + 8 = 0.
- (Standard) Find the nature of the roots of 4x² + 7x − 3 = 0.
- (Standard) Find the nature of the roots of 1x² + 0x + 1 = 0.
- (Challenge) Find k so that 2x² + kx + 18 = 0 has equal roots.
- (Challenge) Find k so that 3x² + kx + 27 = 0 has equal roots.
Answer key with full working
- Q1: D = (-4)² − 4(4)(-1) = 32 → Two distinct real roots → Answer: Two distinct real roots
- Q2: D = (-4)² − 4(1)(8) = -16 → No real roots → Answer: No real roots
- Q3: D = (7)² − 4(4)(-3) = 97 → Two distinct real roots → Answer: Two distinct real roots
- Q4: D = (0)² − 4(1)(1) = -4 → No real roots → Answer: No real roots
- Q5: D = k² − 4(2)(18) = 0 → k² = 144, k = ±12 → Answer: k = ±12
- Q6: D = k² − 4(3)(27) = 0 → k² = 324, k = ±18 → Answer: k = ±18
Syllabus note
This skill is listed under "Quadratic Equations" for Class 10 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is nature of roots using the discriminant in?
"Quadratic Equations", Class 10 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.