Without solving, examine the nature of the roots of the equations 2x² - 6x + 3 = 0.

We need to examine the nature of the roots of the equations 2x² – 6x + 3 = 0.

This can be done by finding the discriminant of the equation.

2x² – 6x + 3 = 0

Comparing the given equation with general form of equation: ax² + bx + c = 0, we get;

a = 2, b = – 6, c = 3

b² – 4ac = (- 6)² – 4(2)(3)

= 36 – 24

= 12

b² – 4ac > 0

Thus, the equation has two distinct real roots.

Since,

x = [- b ± √ (b² – 4ac)] / 2a

x = [-(- 6) ± √12] / (2)2

= (6 ± 2√3) / 4

= (3 ± √3) / 2

Roots of the given equation are x = (3 + √3) / 2 and x = (3 – √3) / 2