Refined and Comprehensive Class 10 Mathematics Notes
Standard Form & Core Concepts
A quadratic equation in the variable x is an equation of the form:
ax² + bx + c = 0, where a ≠ 0
Quadratic Formula
The roots of the quadratic equation ax² + bx + c = 0 are given by:
x = (-b ± √(b² - 4ac)) / 2a
Discriminant & Nature of Roots
The expression D = b² - 4ac is called the discriminant. It determines the nature of the roots:
Discriminant Value
Nature of Roots
D > 0
Two distinct real roots
D = 0
Two equal real roots
D < 0
No real roots
Methods of Solving Quadratic Equations
Factorisation: Splitting the middle term to factorise the quadratic polynomial into linear factors.
Completing the Square: Converting the equation into the form (x + k)² = p.
Quadratic Formula: Directly applying the standard formula using coefficients a, b, and c.
Points to Remember
Condition for Real Roots: A quadratic equation has real roots if D ≥ 0 (i.e., b² - 4ac ≥ 0).
Real-Life Applications: Quadratic equations are widely used to solve practical problems involving area, speed, distance, time, and consecutive integers.
Important Practice Questions
Solve: 3x² - 5x - 2 = 0
Find the value of k for which kx² + 4x + 1 = 0 has equal roots.
Solve a real-life problem involving area, speed, or consecutive integers.