 Nature of Roots Based on Discriminant
 two distinct real roots, two equal real roots, no real roots
 Solutions of Quadratic Equations by Using Quadratic Formula and Nature of Roots

Two distinct real roots if b^{2} – 4ac > 0

Two equal real roots if b^{2} – 4ac = 0

No real roots if b^{2} – 4ac < 0
have seen that the roots of the equation ax^{2} + bx + c = 0 are given by
`x=(b+sqrt(b^24ac))/(2a)`
If b^{2} – 4ac > 0, we get two distinct real roots, `b/(2a)+(b^24ac)/(2a)` and `d/(2a)(sqrt(b^24ac))/(2a)`
If b^{2} – 4ac = 0,then x= `b/(2a)+0` i.e., `x=b/(2a) or b/(2a)`
So, the roots of the equation ax^{2} + bx + c = 0 are both `b/(2a)`
Therefore, we say that the quadratic equation ax^{2} + bx + c = 0 has two equal real roots in this case.
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.
Since b^{2} – 4ac determines whether the quadratic equation ax^{2} + bx + c = 0 has real roots or not, b2 – 4ac is called the discriminant of this quadratic equation.
So, a quadratic equation ax^{2} + bx + c = 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 one examples.
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.