Last Updated: May 2026
CBSE Class 10 Maths Chapter 4 — Quadratic Equations is a 12-mark chapter in the Class 10 Maths board paper (one MCQ + one Short Answer + one Long Answer typically). The chapter introduces the quadratic equation ax² + bx + c = 0, three methods of solution (factorisation, completing the square, quadratic formula), the discriminant, and applied word problems. This 1,800-word CBSE Class 10 Quadratic Equations guide covers all NCERT exercises (4.1–4.4), solved examples and 25 important questions with full working.
1. Definition of Quadratic Equation
A quadratic equation is a polynomial equation of degree 2 of the form:
ax² + bx + c = 0, where a ≠ 0.
The values of x that satisfy this equation are called roots or zeros.
2. Method 1 — Factorisation
Split the middle term: find two numbers whose product = a·c and sum = b.
Example: x² + 5x + 6 = 0. Numbers: 2 and 3 (sum 5, product 6). x² + 2x + 3x + 6 = 0 → x(x+2) + 3(x+2) = 0 → (x+2)(x+3) = 0. Roots: x = -2, -3.
3. Method 2 — Completing the Square
Convert ax² + bx + c = 0 to (x + b/2a)² = (b² – 4ac)/(4a²).
Example: x² + 4x – 5 = 0. (x+2)² = 9 → x+2 = ±3 → x = 1 or -5.
4. Method 3 — Quadratic Formula
x = [-b ± √(b² – 4ac)] / (2a)
This works for ALL quadratic equations.
5. Discriminant (D)
D = b² – 4ac
| Discriminant | Nature of Roots |
|---|---|
| D > 0 | Two distinct real roots |
| D = 0 | Two equal real roots (= -b/2a) |
| D < 0 | No real roots (complex conjugate pair) |
6. Sum and Product of Roots (Vieta’s)
If α and β are roots of ax² + bx + c = 0:
α + β = -b/a
α · β = c/a
Example: Roots of 2x² – 5x + 3 = 0: sum = 5/2, product = 3/2.
7. Forming Quadratic from Given Roots
If roots are α and β, equation = x² – (α+β)x + α·β = 0.
Example: Roots 4 and -2 → x² – 2x – 8 = 0.
8. Real-Life Applications (Word Problems)
| Type | Setup |
|---|---|
| Speed-Distance-Time | Time = Distance/Speed; one variable in distance and another in speed |
| Age problems | Present age vs past/future relations |
| Number problems | ‘Sum of squares of two consecutive numbers’ |
| Rectangular field/area | Length × breadth = area; perimeter constraint |
| Geometry (Pythagoras) | Hypotenuse² = side1² + side2² |
9. Worked Example — Word Problem (Speed)
A train travels 360 km at uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less. Find the speed.
Let speed = x km/h. Time at x = 360/x. Time at x+5 = 360/(x+5).
360/x – 360/(x+5) = 1
360(x+5) – 360x = x(x+5)
1800 = x² + 5x
x² + 5x – 1800 = 0
x = [-5 ± √(25 + 7200)]/2 = [-5 ± 85]/2
x = 40 km/h (rejecting -45)
10. Worked Example — Geometry
The hypotenuse of a right triangle is 13 cm. The other two sides differ by 7 cm. Find the sides.
Let smaller side = x, other = x + 7. Then:
x² + (x+7)² = 13² = 169
2x² + 14x + 49 = 169
2x² + 14x – 120 = 0
x² + 7x – 60 = 0
x = [-7 ± √(49 + 240)]/2 = [-7 ± 17]/2
x = 5 (rejecting -12). Sides: 5, 12, 13.
11. 25 Important Questions (Sample 12)
- Solve 2x² – 7x + 3 = 0 by factorisation. — x = 3 or 1/2
- Find the discriminant of 4x² – 6x + 9 = 0 and nature of roots. — D = 36 – 144 = -108; no real roots
- Solve x² – 4x – 5 = 0 by completing the square. — (x-2)² = 9; x = 5 or -1
- Find roots of x² + x – 12 = 0 by quadratic formula. — x = 3 or -4
- Find k if x² + 2(k+1)x + k² = 0 has equal roots. — D = 0: 4(k+1)² – 4k² = 0; k = -1/2
- Form a quadratic equation whose roots are 2 + √3 and 2 – √3. — Sum = 4, Product = 1; x² – 4x + 1 = 0
- Sum of two numbers is 27 and product 182. Find them. — Roots of t² – 27t + 182 = 0; t = 13 or 14
- The area of a rectangle is 28 cm². Length is 3 cm more than breadth. Find dimensions. — x(x+3)=28; x=4, length=7
- For what value of p, the equation 3x² + 12x + p = 0 has equal roots? — D = 144-12p = 0; p = 12
- If one root of 2x² + kx – 6 = 0 is 2, find k. — 8 + 2k – 6 = 0; k = -1
- Find the sum and product of roots of 5x² + 3x – 2 = 0. — Sum = -3/5, Product = -2/5
- The product of two consecutive positive integers is 306. Find them. — x(x+1) = 306; x=17, integers are 17, 18
12. Common Mistakes
- Forgetting to verify rejected root makes physical sense (negative time, negative dimension).
- Sign errors in completing the square method.
- Discriminant: confusing D > 0 with D < 0.
- Splitting the middle term: choosing wrong factors.
- Quadratic formula: forgetting the ± sign.
13. Tips for CBSE Boards
- Always verify your roots by substitution.
- For word problems, define variable clearly and write ‘let’ statement.
- Show all 4 steps in completing the square (don’t skip).
- State ‘reject’ negative root with reason in word problems.
- For discriminant questions, write D = b² – 4ac explicitly before substituting.
Frequently Asked Questions
Q1. Class 10 board weightage for this chapter?
10–12 marks. Top-3 chapters in CBSE Class 10 Maths.
Q2. Best method to solve quadratics in exams?
Try factorisation first (fastest). If middle term doesn’t split easily, use quadratic formula.
Q3. Will the formula be given in the exam?
No. Memorise quadratic formula AND discriminant formula.
Q4. Can quadratic equations have one solution?
Yes — when D = 0, both roots are equal (i.e., one repeated root).
Internal Resources
- CBSE Class 10 Maths Triangles
- CBSE Class 10 Acids, Bases and Salts
- Board Exam FAQ
- Board Exam Mock Test
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