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

(i) 2x² – 3x + 5 = 0

Comparing this equation with ax2 + bx + c = 0, we obtain,
 a = 2, b = −3, c = 5

Discriminant = b² − 4ac = (− 3)² − 4 (2) (5) = 9 − 40 = −31

As,  b² − 4ac < 0,

Therefore, no real root is possible for the given equation.

(ii)

Comparing this equation with ax² + bx + c = 0,
we obtain,
a = 3
b = -4√3
c = 4

=Discriminant  = 48 − 48 = 0

As,  b² − 4ac = 0,

Therefore, real roots exist for the given equation and they are equal to each other.

And the roots will be  and .

Therefore, the roots are and .

(iii)    2x² – 6x + 3 = 0

Comparing this equation with ax²+bx + c = 0,
we obtain,
 a = 2, b = −6, c = 3

Discriminant = b² − 4ac = (− 6)² − 4 (2) (3) = 36 − 24 = 12

As,  b² − 4ac > 0,

Therefore, distinct real roots exist for this equation as follows.

So, the roots are or .