Quadratic Equations
Factorisation, quadratic formula, discriminant, and word problems.
Chapter Overview
Always write the equation in standard form ax² + bx + c = 0 before applying the quadratic formula.
A quadratic equation ax² + bx + c = 0 (a ≠ 0) has degree 2. Class 10 covers factorisation, completing the square, the quadratic formula, discriminant, and word problems — a high-frequency board topic.
Standard Form and Roots
A quadratic equation in variable x has the form ax² + bx + c = 0, where a ≠ 0. The highest power of x is 2. Solutions are called roots or zeroes of the equation.
Methods of Solving
- Factorisation — split middle term or use identities
- Completing the square
- Quadratic formula: x = (−b ± √(b²−4ac)) / 2a
- Nature of roots from discriminant D = b² − 4ac
Discriminant
- D > 0: two distinct real roots
- D = 0: two equal real roots (repeated root)
- D < 0: no real roots
Solved Examples
Step-by-step solutions — read each step before checking the final answer.
Example 1: Factorisation
Solve x² − 5x + 6 = 0 by factorisation.
- 1Find two numbers: product +6, sum −5 → (−2) and (−3).
- 2x² − 5x + 6 = (x − 2)(x − 3).
- 3x − 2 = 0 → x = 2; x − 3 = 0 → x = 3.
- 4Verify: (2)² − 5(2) + 6 = 0 ✓
Answer
x = 2 or x = 3
Example 2: Word Problem
The product of two consecutive positive integers is 306. Find the integers.
- 1Let integers be n and (n + 1).
- 2n(n + 1) = 306 → n² + n − 306 = 0.
- 3Factorise: (n + 18)(n − 17) = 0 → n = 17 (positive).
- 4Integers are 17 and 18.
Answer
17 and 18
Key Points to Remember
- ✓ Always write equation as ax² + bx + c = 0 first
- ✓ Verify roots by substitution
- ✓ Word problems → form equation → solve
- ✓ Sum of roots = −b/a; product = c/a
Exam Tips
- • Show factorisation steps for full marks
- • State discriminant when asked about nature of roots
- • Board exams often combine quadratics with word problems
Practice MCQs — Quadratic Equations
Test your understanding with topic-wise multiple choice questions. Explanations appear after each answer.
Roots of x² − 9 = 0 are: