**Basic Concepts:** An equation of the form ax^{2} + bx + c = 0, where a ≠0 and a, b, c are real numbers, is called a quadratic equation.

The numbers a, b, c are called the coefficients of the quadratic equation.

A root of the quadratic equation is a number α (real or complex) such that aα^{2} + bα + c = 0.

The roots of the quadratic equation are given by

x =

The quantity D (D= b^{2} – 4ac) is known as the **discriminant** of the equation

__Basic Results:__

**(i)** The quadratic equation has real and equal roots if and only if D = 0

i.e. b^{2} – 4ac = 0.

**(ii)** The quadratic equation has real and distinct roots if and only if D > 0

i.e. b^{2} – 4ac > 0.

**(iii)** The quadratic equation has complex roots with non-zero imaginary parts if and only if D < 0

i.e. b^{2} – 4ac < 0.

**(iv)** If p + iq (p and q being real) is a root of the quadratic equation where i = √-1 , then p – iq is also a root of the quadratic equation.

**(v)** If p + √q is an irrational root of the quadratic equation, then p – √q is also a root of the quadratic equation provided that all the coefficients are rational , q not being a perfect square or zero.

**(vi)** The quadratic equation has rational roots if D is a perfect square and a, b, c are rational.

**(vii)** If a = 1 and b, c are integers and the roots of the quadratic equation are rational, then the roots must be integers.

**(viii)** If the quadratic equation is satisfied by more than two distinct numbers (real or complex), then it becomes an identity i.e. a = b = c = 0.

**(ix)** Let α and β be two roots of the given quadratic equation. Then α + β = -b/a and αβ = c/a .

**(x)** A quadratic equation, whose roots are α and β can be written as

(x – α) (x – β) = 0

i.e., ax^{2} + bx + c ≡ a(x – α) (x – β).