We have seen that the roots of the equation ${ax^2 + bx + c = 0}$ are given by
$$x={-b±√{b^2-4ac}}/{2a}$$
where ${b^2 – 4ac}$ is called the discriminant of this quadratic equation.

Case 1: If $b^2 – 4ac > 0$, we get two distinct real roots,

$x={-b+√{b^2-4ac}}/{2a}$ and $x={-b-√{b^2-4ac}}/{2a}$

Case 2: If $b^2 – 4ac = 0$, then has two equal real roots in this case.

$x=-b/{2a}$

Case 3:If ${b^2 – 4ac < 0}$, then there is no real number whose square is $b^2 – 4ac$. Therefore, there are no real roots for the given quadratic equation in this case.

Example: Find the discriminant of the quadratic equation ${2x^2 – 4x + 3 = 0}$, and hence find the nature of its roots.
Solution: The given equation is of the form ${ax^2 + bx + c = 0}$, where $a = 2$, $b = – 4$ and $c = 3$. Therefore, the discriminant
$b^2 – 4ac$ = (– 4)² – (4 × 2 × 3) = 16 – 24 = – 8 < 0
So, the given equation has no real roots.

Homework:A pole has to be erected at a point on the boundary of a circular park of diameter 13 metres in such a way that the differences of its distances from two diametrically opposite fixed gates Aand B on the boundary is 7 metres. Is it possible to do so? If yes, at what distances from the two gates should the pole be erected?

Homework:Find the discriminant of the equation ${3x^2 – 2x + 1/3 = 0}$ and hence find the nature of its roots. Find them, if they are real.

EXERCISE

1.   Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:

(i)   ${2x^2 – 3x + 5 = 0 }$                           (ii)  ${3x^2 –   4   3 x + 4 = 0}$ 
(iii)   ${2x^2 – 6x + 3 = 0}$

2.   Find the values of $k$ for each of the following quadratic equations, so that they have two equal roots.
(i)   ${2x^2 + kx + 3 = 0}$                                       (ii)  ${kx (x – 2) + 6 = 0}$

3.   Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 $m^2$? 
     If so, find its length and breadth.
	 
4.   Is the following situation possible? If so, determine their present ages.
     The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.
	 
5.   Is it possible to design a rectangular park of perimeter 80$m$ and area 400 $m^2$? If  so, find its length and breadth.