Quadratic Equations
Chapter 4 | Mathematics
Complete NCERT notes, standard form, methods (factorization, completing square, quadratic formula), discriminant, nature of roots, word problems, and key points for CBSE Class 10 Board Exam 2025-26.
📖 Introduction
A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.
The term "quadratic" comes from "quadratum" (Latin for square) because the variable gets squared. These equations arise in many real-life situations: projectile motion, area problems, profit maximization, etc.
📐 Standard Form of Quadratic Equation
General form: ax² + bx + c = 0
- a = coefficient of x² (must be non-zero)
- b = coefficient of x
- c = constant term
✅ Examples
2x² + 3x - 5 = 0 → a=2, b=3, c=-5
x² - 4x = 0 → a=1, b=-4, c=0
3x² + 2 = 0 → a=3, b=0, c=2
💡 If a = 0, the equation becomes linear (bx + c = 0), not quadratic.
✖️ Factorization Method (Splitting the Middle Term)
Steps:
- Multiply a and c (ac).
- Find two numbers whose product = ac and sum = b.
- Split the middle term using these numbers.
- Factor by grouping and solve for x.
📝 Example
Solve: x² - 5x + 6 = 0
ac = 6, b = -5 → numbers -2 and -3 (product 6, sum -5)
x² - 2x - 3x + 6 = 0 → x(x-2) -3(x-2)=0 → (x-2)(x-3)=0
→ x = 2 or x = 3
🔲 Completing the Square Method
Steps:
- Write equation as ax² + bx = -c (move constant to RHS).
- Divide by a if a ≠ 1.
- Add (b/2a)² to both sides to complete the square.
- Write LHS as perfect square (x + b/2a)².
- Take square root and solve.
📝 Example
Solve: x² + 4x - 5 = 0
x² + 4x = 5
Add (4/2)² = 4 to both sides: x² + 4x + 4 = 9
(x + 2)² = 9 → x + 2 = ±3 → x = 1 or x = -5
📏 Quadratic Formula (Sridharacharya Formula)
This formula gives the roots of any quadratic equation ax² + bx + c = 0, a ≠ 0.
📝 Example
Solve: 2x² - 7x + 3 = 0
a=2, b=-7, c=3
x = [7 ± √(49 - 24)] / 4 = [7 ± √25]/4 = [7 ± 5]/4
x = 12/4 = 3 or x = 2/4 = 0.5
🎯 Discriminant (D)
The discriminant determines the nature of the roots without solving the equation.
🌱 Nature of Roots
| 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, not in Class 10) |
💡 Additional: If D is a perfect square, roots are rational; otherwise, they are irrational.
📝 Sum and Product of Roots
For ax² + bx + c = 0,
Sum of roots (α+β) = -b/a
Product of roots (αβ) = c/a
📝 NCERT Solved Examples (In-depth)
Example 1: Factorization Method
Solve: 2x² - 5x + 3 = 0
ac = 6, b = -5 → numbers -2 and -3
2x² - 2x - 3x + 3 = 0 → 2x(x-1) -3(x-1)=0 → (x-1)(2x-3)=0
→ x = 1 or x = 3/2
Example 2: Completing the Square
Solve: 4x² + 4√3x + 3 = 0
Divide by 4: x² + √3x + 3/4 = 0 → x² + √3x = -3/4
Add (√3/2)² = 3/4: x² + √3x + 3/4 = 0 → (x + √3/2)² = 0 → x = -√3/2 (repeated root)
Example 3: Quadratic Formula
Solve: 3x² - 2√6x + 2 = 0
a=3, b=-2√6, c=2
D = (-2√6)² - 4×3×2 = 24 - 24 = 0
x = (2√6 ± 0)/(6) = √6/3 (repeated root)
Example 4: Word Problem (Area)
The area of a rectangular plot is 528 m². Length is one more than twice its breadth. Find length and breadth.
Let breadth = x m, length = (2x+1) m
x(2x+1) = 528 → 2x² + x - 528 = 0
D = 1 + 4224 = 4225 = 65²
x = (-1 ± 65)/4 → x = 64/4=16 or x = -66/4 (reject)
Breadth = 16 m, Length = 33 m
Example 5: Nature of Roots
Find k so that 2x² + kx + 3 = 0 has equal roots.
For equal roots, D = 0 → k² - 4×2×3 = 0 → k² - 24 = 0 → k = ±√24 = ±2√6
⭐ Key Points for Board Exam (Deep Research)
- Quadratic equation: ax² + bx + c = 0, a ≠ 0.
- Roots: Values of x that satisfy the equation.
- Methods to solve: Factorization, completing square, quadratic formula.
- Discriminant (D): D = b² - 4ac → determines nature of roots.
- Nature of roots: D>0 → two distinct real; D=0 → equal real; D<0 → no real roots.
- Sum of roots (α+β) = -b/a; Product (αβ) = c/a.
- Form quadratic from roots: x² - (α+β)x + αβ = 0.
- If a and c have opposite signs → D always positive → two distinct real roots.
- For rational roots, D must be a perfect square and a,b,c rational.
- Word problems: Numbers, ages, area, speed, time, profit, etc. — always reject extraneous roots.