1.   A quadratic equation in the variable $x$ is of the form ${ax^2 + bx + c = 0}$, 
where $a$, $b$,$c$ are real numbers and $a ≠ 0$.

2.   A real number $α $ is said to be a root of the quadratic equation ${ax^2 + bx + c = 0}$, 
if $aα^2 + bα + c = 0$. The zeroes of the quadratic polynomial ${ax^2 + bx + c = 0}$ and the roots 
of the quadratic equation ${ax^2 + bx + c = 0}$ are the same.

3.   If we can factorise  ${ax^2 + bx + c = 0}$, into a product of two linear factors, then the roots
 of the quadratic equation  ${ax^2 + bx + c = 0}$ can be found by equating each factor to zero.
 
4.   A quadratic equation can also be solved by the method of completing the square.
5.   Quadratic formula: The roots of a quadratic equation ${ax^2 + bx + c = 0}$ are given by
${-b±√{b^2-4ac}}/{2a}$,provided $b^2 – 4ac ≥ 0$.

6.   A quadratic equation ${ax^2 + bx + c = 0}$ has
(i)   two distinct real roots, if $b^2 – 4ac ≥ 0$.,
(ii)   two equal roots (i.e., coincident roots), if $b^2 – 4ac = 0$., and
(iii)   no real roots, if $b^2  – 4ac < 0$. 

7. Important:Relative Speed. When two bodies move in opposite direction, then their 
    relative speed = $v_1 + v_2$.
	When two bodies move in same direction, then their 
    relative speed = $v_1 - v_2$.
	
	Where $v_1$ and $v_2$ are the speed of bodies.
	
8. Important:Relative Speed. If the spped of the boat in still water be $x$ km/h
	and the speed of stream by $y$ km/h, then the relative speed of the boat is as follows:
	
	Speed of boat in upstream direction (i.e opposite to the flow of stream)= $(x-y)$ km/hr
	
	Speed of boat in downstream direction (i.e along then direction of flow of stream)= $(x+y)$ km/hr