Quadratic Equations — Mathematics Class 10 Notes (CBSE & HBSE)
Free NCERT Mathematics notes for Quadratic Equations (Class 10) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 3 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Quadratic Equations (CBSE & HBSE)
High-priority chapter in both boards. CBSE tests factorisation, completing the square, discriminant, and word problems. HBSE frequently asks nature of roots and nature of solutions problems.
Standard Form and Factorisation Method
Standard Form of Quadratic Equation
ax² + bx + c = 0 where a ≠ 0
- a = coefficient of x² (must be non-zero)
- b = coefficient of x
- c = constant term
Roots of Quadratic Equation
A value x = α is a root if it satisfies the equation (i.e., aα² + bα + c = 0).
A quadratic equation has at most 2 roots.
Factorisation Method (Splitting the Middle Term)
Step 1: Find two numbers p, q such that p + q = b and p × q = ac Step 2: Rewrite bx = px + qx Step 3: Factor by grouping Step 4: Set each factor = 0 and solve
Example: x² - 5x + 6 = 0
- ac = 1×6 = 6; p+q = -5; p×q = 6
- p = -3, q = -2
- x² - 3x - 2x + 6 = x(x-3) - 2(x-3) = (x-2)(x-3)
- Roots: x = 2 and x = 3
Useful Factorisation Identities
| Identity | Form |
|---|---|
| Difference of squares | a² - b² = (a+b)(a-b) |
| Perfect square | a² ± 2ab + b² = (a ± b)² |
Check: Substitute roots back. For x=2: 4-10+6=0 ✓; For x=3: 9-15+6=0 ✓
Quadratic Formula and Completing the Square
Completing the Square Method
Standard quadratic: ax² + bx + c = 0 Steps:
- Divide by a: x² + (b/a)x + (c/a) = 0
- Add & subtract (b/2a)²:
- Solve for x
[x + b/(2a)]² = b²/(4a²) - c/a
Example: x² - 4x - 5 = 0
- (x-2)² - 4 - 5 = 0 → (x-2)² = 9
- x-2 = ±3 → x = 5 or x = -1
The Quadratic Formula
Derived from completing the square: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
The Discriminant: D = b² - 4ac
| Discriminant D | Nature of Roots |
|---|---|
| D > 0 | Two distinct real roots |
| D = 0 | Two equal real roots (one repeated root) |
| D < 0 | No real roots (complex roots) |
Example: x² - 7x + 10 = 0
- a=1, b=-7, c=10
- D = 49 - 40 = 9 > 0 → Two distinct real roots
- x = (7 ± 3)/2 → x = 5 or x = 2
When to Use Which Method?
| Method | Best When |
|---|---|
| Factorisation | Factors are integers/simple fractions |
| Completing the square | Asked explicitly; or proving formula |
| Quadratic formula | Always works; use for complex/irrational roots |
CBSE Tip: Discriminant (D) value determines nature of roots — very frequently tested as 1-mark questions.
Word Problems on Quadratic Equations
Setting Up Quadratic Word Problems
- Read carefully — identify what x represents
- Form the quadratic equation from conditions
- Solve (factorisation or formula)
- Reject negative/impossible roots based on context
- Answer the question (the root may not be the final answer!)
Common Problem Types
Type 1: Consecutive Numbers "Product of two consecutive positive integers is 182."
- Let integers = n and n+1
- n(n+1) = 182 → n² + n - 182 = 0
- (n-13)(n+14) = 0 → n = 13 (reject -14)
- Integers: 13 and 14
Type 2: Speed/Train Problems "A train travels 360 km at speed v. If speed increased by 5, journey 1 hr shorter."
- 360/v - 360/(v+5) = 1 → 360(v+5) - 360v = v(v+5)
- 1800 = v² + 5v → v² + 5v - 1800 = 0
Type 3: Area Problems "Rectangle perimeter=48 m, area=108 m². Find dimensions."
- Let l = length, l + w = 24, lw = 108
- l² - 24l + 108 = 0 → (l-6)(l-18) = 0
- l = 18 m, w = 6 m
Type 4: Work Problems "Two pipes fill tank in 6 hours together. Pipe A alone takes 5 hours more than B."
- Let B takes t hours, A takes t+5 hours
- 1/t + 1/(t+5) = 1/6 → 6(2t+5) = t(t+5) → t²-7t-30=0
Key Rule: Always check if the solution is practically valid (no negative time, negative length, etc.)
Frequently asked questions
Are these Quadratic Equations notes free?
Yes — the Quadratic Equations notes for Mathematics (Class 10) on Siksha Sarovar are completely free to read, with no account required.
Do these notes follow CBSE and HBSE?
Yes. The Quadratic Equations notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.
What does the Quadratic Equations chapter cover?
Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Quadratic Equations.