Chapter 4: Quadratic Equations
Mastering factorisation, the quadratic formula, and the nature of roots.
📘 Standard Form & Roots
Any equation of degree 2 is a quadratic equation.
Examples: 2x² + x - 300 = 0.
A real number α is a root if aα² + bα + c = 0. The zeroes of the quadratic polynomial and roots of the equation are the same.
⚙️ Solving by Factorisation
Factorise ax² + bx + c into a product of two linear factors, then equate each factor to zero.
2x² - 2x - 3x + 3 = 0
(2x - 3)(x - 1) = 0
Roots: x = 3/2 or x = 1.
📈 Nature of Roots & The Discriminant
The expression b² - 4ac is called the discriminant because it determines whether the quadratic equation has real roots or not.
Two Distinct Real Roots
Two Equal Real Roots
(Coincident Roots)
No Real Roots
🔗 The Quadratic Formula
If b² - 4ac ≥ 0, the roots of ax² + bx + c = 0 are given by:
(Historically derived by Sridharacharya)
⭐ Board Exam Hacks
Checking for Quadratics
Don't judge by appearances! Expand equations fully. (x+2)³ = x³ - 4 looks cubic, but x³ cancels out leaving 6x² + 12x + 12 = 0, making it a quadratic equation!
⚠️ Common Traps
The "Real Root" Condition
If a question says "Find roots if they are real," always calculate b² - 4ac FIRST. If it's negative, simply write "No real roots" and save time calculating further!