Zeroes and coefficients of a quadratic: Class 10 practice
For ax² + bx + c with zeroes α and β: α + β = −b/a and αβ = c/a. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Factorising quadratics
The idea behind zeroes and coefficients of a quadratic
For ax² + bx + c with zeroes α and β: α + β = −b/a and αβ = c/a. A quadratic with given zeroes is k[x² − (α + β)x + αβ].
Step-by-step method
- Write sum S and product P of zeroes.
- Form x² − Sx + P.
- Multiply by a constant to clear fractions if needed.
Worked example 1 (Basic)
Find a quadratic polynomial with sum of zeroes -2 and product 1.
- x² − (-2)x + (1)
- Answer: x² + 2x + 1
Worked example 2 (Standard)
Find the zeroes of x² + 3x − 4 and verify the relations.
- Factor: (x + 4)(x − 1)
- Zeroes -4, 1; sum -3 = −(3)/1 ✓, product -4 ✓
- Answer: -4 and 1
Worked example 3 (Challenge)
Find a quadratic whose zeroes have sum 5/4 and product −1/4.
- x² − (5/4)x − 1/4
- Multiply by 4: 4x² − 5x − 1
- Answer: 4x² − 5x − 1
Common mistakes and misconceptions
- Sign error: using +S
- Confusing −b/a with b/a
Practice questions
- (Basic) Find a quadratic polynomial with sum of zeroes -2 and product -15.
- (Basic) Find a quadratic polynomial with sum of zeroes -3 and product -10.
- (Standard) Find the zeroes of x² + 0x − 9 and verify the relations.
- (Standard) Find the zeroes of x² + 2x − 24 and verify the relations.
- (Challenge) Find a quadratic whose zeroes have sum 3/2 and product −1/2.
- (Challenge) Find a quadratic whose zeroes have sum 2/2 and product −1/2.
Answer key with full working
- Q1: x² − (-2)x + (-15) → Answer: x² + 2x − 15
- Q2: x² − (-3)x + (-10) → Answer: x² + 3x − 10
- Q3: Factor: (x − 3)(x + 3) → Zeroes 3, -3; sum 0 = −(0)/1 ✓, product -9 ✓ → Answer: 3 and -3
- Q4: Factor: (x − 4)(x + 6) → Zeroes 4, -6; sum -2 = −(2)/1 ✓, product -24 ✓ → Answer: 4 and -6
- Q5: x² − (3/2)x − 1/2 → Multiply by 2: 2x² − 3x − 1 → Answer: 2x² − 3x − 1
- Q6: x² − (2/2)x − 1/2 → Multiply by 2: 2x² − 2x − 1 → Answer: 2x² − 2x − 1
Syllabus note
This skill is listed under "Polynomials" 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 zeroes and coefficients of a quadratic in?
"Polynomials", 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.