**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**

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

__Basic Results:__

__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).**