Basic Concepts: An equation of the form ax2 + 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 aα2 + bα + c = 0.

The roots of the quadratic equation are given by $\displaystyle x = \frac{-b \pm \sqrt{b^2 - 4a c}}{2 a}$

The quantity D (D = b2 – 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. b2 – 4ac = 0.

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

i.e. b2 – 4ac > 0.

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

i.e. b2 – 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., ax2 + bx + c ≡ a(x – α) (x – β).

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