Last Updated: April 2026 | CBSE Class 10 Mathematics | NCERT Chapter 4 Notes & MCQs
Quadratic equations are one of the most rewarding chapters in CBSE Class 10 Mathematics — they carry approximately 10 marks in the board exam (one 1-mark MCQ + one 2-mark + one 3-mark + occasionally one 5-mark word problem). Master the four standard methods, the discriminant rule, and word-problem framing, and you will routinely score full marks here. This guide gives you complete NCERT Chapter 4 notes, every formula you need, three solved examples, an exercise-by-exercise outline, and a 10-MCQ embedded quiz at the end.
1. What is a Quadratic Equation?
A polynomial equation of degree 2 in a single variable x is called a quadratic equation. Its standard form is:
ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.
Here a is the coefficient of x², b is the coefficient of x, and c is the constant term. The condition a ≠ 0 is critical — if a = 0, the equation reduces to a linear equation.
Examples of quadratic equations:
- 2x² − 5x + 3 = 0 (a = 2, b = −5, c = 3)
- x² − 9 = 0 (a = 1, b = 0, c = −9)
- 3x(x + 2) = 5 → 3x² + 6x − 5 = 0
Non-examples: x³ + 2x = 5 (degree 3), 1/x + x = 2 (not a polynomial), √x + 3 = 0 (not a polynomial).
2. Roots of a Quadratic Equation
A real number α is called a root (or solution, or zero) of the quadratic equation ax² + bx + c = 0 if aα² + bα + c = 0. Geometrically, the roots are the x-intercepts of the parabola y = ax² + bx + c. A quadratic equation can have at most two roots.
3. Methods of Solving Quadratic Equations (NCERT)
Method 1 — Factorisation by Splitting the Middle Term
Steps for ax² + bx + c = 0:
- Compute the product a × c.
- Find two numbers p and q such that p + q = b and p × q = ac.
- Rewrite bx as px + qx and factor by grouping.
- Set each factor to zero to find the roots.
Example: Solve 6x² − x − 2 = 0.
ac = −12. Numbers: −4 and 3 (sum = −1, product = −12).
6x² − 4x + 3x − 2 = 0 → 2x(3x − 2) + 1(3x − 2) = 0 → (2x + 1)(3x − 2) = 0.
Roots: x = −1/2, x = 2/3.
Method 2 — Completing the Square
Convert ax² + bx + c = 0 into the form (x + b/2a)² = (b² − 4ac)/4a², then take square roots.
Example: Solve x² + 4x − 5 = 0.
x² + 4x = 5 → (x + 2)² = 9 → x + 2 = ±3 → x = 1 or x = −5.
Method 3 — Quadratic Formula (Sridharacharya’s Formula)
For ax² + bx + c = 0, provided b² − 4ac ≥ 0:
x = [ −b ± √(b² − 4ac) ] / 2a
Example: Solve 2x² − 7x + 3 = 0.
D = 49 − 24 = 25, √D = 5. x = (7 ± 5)/4 = 3 or 1/2. Roots: 3, 1/2.
Method 4 — Graphical Method (Conceptual Only — Not Required for Solving)
The roots correspond to the x-intercepts of the parabola. If the parabola does not cross the x-axis, no real roots exist.
4. The Discriminant — Nature of Roots
The expression D = b² − 4ac is called the discriminant. It tells us the nature of the roots without solving the equation.
| Discriminant (D) | Nature of Roots | Geometric Meaning |
|---|---|---|
| D > 0 | Two distinct real roots | Parabola cuts x-axis at 2 points |
| D = 0 | Two equal real roots (coincident) | Parabola touches x-axis at 1 point |
| D < 0 | No real roots (roots are imaginary) | Parabola does not meet x-axis |
5. Sum and Product of Roots (Vieta’s Relations)
If α and β are the roots of ax² + bx + c = 0, then:
- Sum of roots: α + β = −b/a
- Product of roots: αβ = c/a
Conversely, a quadratic equation with given roots α, β can be written as:
x² − (α + β)x + αβ = 0
6. CBSE Marks Weightage
| Question Type | Marks | Typical Topic |
|---|---|---|
| MCQ / Assertion-Reason | 1 | Discriminant, nature of roots |
| Short Answer (SA-I) | 2 | Solve by factorisation / formula |
| Short Answer (SA-II) | 3 | Find k for equal roots, sum/product applications |
| Long Answer (LA) | 5 | Word problem (speed-distance, age, area) |
| Total weightage | ~10–11 | Approx. 12% of Maths paper |
7. NCERT Exercise Solutions Outline (Chapter 4)
- Exercise 4.1 (5 questions) — Identify whether given equations are quadratic; convert real-life situations (Rohan-Reena age problem, Pratima’s marbles, train speed, rectangular plot) into quadratic form.
- Exercise 4.2 (6 questions) — Solve by factorisation: x² − 3x − 10 = 0; 2x² + x − 6 = 0; √2 x² + 7x + 5√2 = 0; word problems on consecutive integers and rectangular field.
- Exercise 4.3 (11 questions) — Use the quadratic formula and discriminant. Find roots and the value of k for which the equation has equal roots. Word problems on time-work, train delay, and Zeba’s age.
- Exercise 4.4 (5 questions) — Determine the nature of roots; if real roots exist, find them. Mostly tests the D > 0 / = 0 / < 0 classification.
8. High-Yield Word Problem Patterns
Pattern A — Speed–Distance–Time: “A train travelling at uniform speed for 360 km would have taken 48 minutes less if its speed were 5 km/h more. Find the original speed.” → Set up D/S = T → quadratic in S.
Pattern B — Age problems: Past or future ages whose product/sum is given.
Pattern C — Geometry / Area: Diagonal, rectangle, right triangle problems where Pythagoras + area condition produce a quadratic.
Pattern D — Time–Work / Pipes: Two pipes filling a tank in given times; find individual rates.
9. Common Mistakes to Avoid
- Forgetting a ≠ 0 when stating the standard form.
- Sign errors in computing D — write b² − 4ac on a separate line.
- Dropping the negative root after taking √D — always write x = (−b ± √D)/2a.
- Reporting a negative value for speed, time, or length in word problems — reject the negative root with a one-line justification.
- Confusing “real and equal” (D = 0) with “no real roots” (D < 0).
10. Quick Revision Formula Sheet
- Standard form: ax² + bx + c = 0, a ≠ 0
- Discriminant: D = b² − 4ac
- Quadratic formula: x = (−b ± √D) / 2a
- Sum of roots: α + β = −b/a
- Product of roots: αβ = c/a
- Equation from roots: x² − (α + β)x + αβ = 0
- Equal roots condition: D = 0
- Real & distinct roots: D > 0
- No real roots: D < 0
11. Practice MCQ Quiz — 10 Questions
Test yourself on Quadratic Equations. Each question carries 1 mark; explanations are revealed after submission.
Quiz data empty after normalization.
12. Frequently Asked Questions
Q1. How many marks does Quadratic Equations carry in the CBSE Class 10 board exam?
Approximately 10–11 marks every year — typically one MCQ (1 mark), one 2-mark factorisation, one 3-mark discriminant question, and one 5-mark word problem.
Q2. Which method is best for the board exam?
For numerical problems, factorisation by splitting the middle term is fastest. If factorisation is not obvious within 30 seconds, switch to the quadratic formula. Completing the square is rarely asked directly but is the derivation of the formula.
Q3. What is the difference between roots and zeroes?
The terms are interchangeable. “Roots of the equation ax² + bx + c = 0” and “zeroes of the polynomial ax² + bx + c” mean the same thing — the values of x that make the expression equal to zero.
Q4. How do I find the value of k for which an equation has equal roots?
Set the discriminant to zero: b² − 4ac = 0, and solve for k. Always remember that k can have two values (e.g., ±4).
Q5. Are imaginary roots part of the Class 10 syllabus?
No — Class 10 only deals with real roots. When D < 0, you simply state “the equation has no real roots.” Imaginary numbers are introduced in Class 11.
13. Continue Your Class 10 Maths Prep
- Class 10 Maths Chapter 2 — Polynomials: Notes & MCQs
- Class 10 Maths Chapter 3 — Pair of Linear Equations: NCERT Notes
- Class 10 Maths Chapter 6 — Triangles: NCERT Solutions & Theorems
- CBSE Sample Paper 2027 with Marking Scheme
- How to Prepare for Class 10 Board Exams — 90-Day Plan
Last Updated: April 2026 | Ready For Boards — your CBSE/ICSE Class 10 & 12 prep partner.
