In this case, the graph neither intersects nor touches the X-axis at all. So, there are no real
roots.
Since b
2
– 4ac determines whether the quadratic equation ax
2
+ bx + c = 0 (a ¹ 0) has
real roots or not, b
2
– 4ac is called the discriminant of the quadratic equation.
So, a quadratic equation ax
2
+ bx + c = 0 (a ¹ 0) has
i. two distinct real roots, if b
2
– 4ac > 0,
ii. two equal real roots, if b
2
– 4ac = 0,
iii. no real roots, if b
2
– 4ac < 0.
Let us consider some examples
Example-14. 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 in the form of 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.
Example-15. 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 A and 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?
Solution : Let us first draw the diagram.
Let P be the required location of the pole. Let the distance of the
pole from the gate B be x m, i.e., BP = x m. Now the difference of the
distances of the pole from the two gates = AP – BP (or, BP – AP)= 7 m.
Therefore, AP = (x + 7) m.
Now, AB = 13m, and since AB is a diameter,
ÐAPB = 900
(Why?)
Therefore, AP2
+ PB2
= AB2
(By Pythagoras theorem)
i.e., (x + 7)2
+ x
2
= 132
i.e., x
2
+ 14x + 49 + x
2
= 169
i.e., 2x
2
+ 14x – 120 = 0
