Key Points - Quadratic Equation

-  Quadratic Polynomial: A polynomial of the form ax^2 + bx + c is called a quadratic expression in the variable x. This is a polynomial of the second degree. In quadratic expression ax^2 + bx + c , a is the coefficient of x^2 , b is the coefficient of x and c is the constant term (or coefficient of x° ..
-  Quadratic Equation: An equation of the form ax^2 + bx + c = 0 , a ≠ 0 , is called a quadratic equation in one variable x, where a, b, c are constants.
-  The equation ax 2 + bx + c = 0, a ≠ 0 is the standard form of a quadratic equation, where a, b and c are real numbers.
-  A real number α is said to be a root of the quadratic equation ax^2 + bx + c = 0, a ≠ 0 .
If aα 2 + bα + c = 0, the zeroes of quadratic polynomial ax^2 + bx + c and the roots of the quadratic equation ax^2 + bx + c = 0 are the same.
-  If we can factorise ax^2 + bx + c = 0, a ≠ 0 into product of two linear factors, then the roots of the quadratic equation can be found by equating each factors to zero.
-  The roots of a quadratic equation ax^2 + bx + c = 0, a ≠ 0 are given by −b ± sqrt(b2 − 4ac) / 2a
provided that b^2 – 4ac ≥ 0.
-  A quadratic equation ax^2 + bx + c = 0, a ≠ 0 has ___________ (a) Two distinct and real roots, if b^2 − 4ac > 0. (b) Two equal and real roots, if b^2 − 4ac = 0. (c) Two roots are not real, if b^2 − 4ac < 0.

-  A quadratic equation can also be solved by the method of completing the square. 
(i) a^2 + 2ab + b^2 = (a+ b)^2
(ii) a^2 - 2ab + b^2 =  ( a -b )^2

Discriminant of the quadratic equation ax 2 + bx + c = 0, a ≠ 0 is given by D = b 2 − 4ac .