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Mathematics: Class 10 — Chapter 4: Quadratic Equations

4.1 Introduction & History
4.2 Quadratic Equations Definition

Standard Form
  • A quadratic equation in the variable x is of the form:
    ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.
  • The standard form requires terms to be written in descending order of their degrees.

Identifying Quadratic Equations
  • An equation must simplify to a polynomial equation of degree 2.
  • Example: (x - 2)² + 1 = 2x - 3 simplifies to x² - 6x + 8 = 0 (Quadratic).
  • Example: x(x + 1) + 8 = (x + 2)(x - 2) simplifies to x + 12 = 0 (Not Quadratic).
4.3 Solution by Factorisation

Roots of a Quadratic Equation
  • A real number α is called a root of the quadratic equation ax² + bx + c = 0 if aα² + bα + c = 0.
  • The roots of the quadratic equation are identical to the zeroes of the corresponding quadratic polynomial.
  • A quadratic equation can have at most two real roots.

Method of Splitting the Middle Term
  • To solve ax² + bx + c = 0, split the middle term bx into two terms such that their product equals ac · x².
  • Factorise the expression into two linear factors and equate each factor to zero.
  • Example: For 2x² - 5x + 3 = 0, split -5x into -2x - 3x.
    (2x - 3)(x - 1) = 0 → Roots are x = 3/2 and x = 1.
4.4 Nature of Roots

The Quadratic Formula
  • The roots of ax² + bx + c = 0 are given by:
    x = [-b ± √(b² - 4ac)] / 2a
  • The term b² - 4ac is called the discriminant (D) of the quadratic equation.

Nature of Roots Based on Discriminant (D = b² - 4ac)
  • If b² - 4ac > 0: Two distinct real roots.
  • If b² - 4ac = 0: Two equal real roots (coincident roots), both equal to -b / 2a.
  • If b² - 4ac < 0: No real roots.
4.5 Summary of Key Concepts