Question

Discuss the nature of the roots of the following equations without actually solving it:

2x² + 2x + 3 = 0

Answer 1

Given: 2x² + 2x + 3 = 0

Step-wise calculation:

1. Compare with ax² + bx + c = 0:

a = 2, b = 2, c = 3

2. Discriminant D = b² – 4ac

= 2² – 4 × 2 × 3 

= 4 – 24

= –20

3. The sign of the discriminant determines the nature of the roots:

D < 0 ⇒ Roots are non‑real (often called imaginary);

D = 0 ⇒ Equal real roots;

D > 0 ⇒ Distinct real roots.

Here, D = –20 which is negative and not zero, so the equation has two non‑real (imaginary) roots and they are distinct (unequal). 

Equivalently, the roots have the general form `(-b ± sqrt(D))/(2a)` so the plus and minus give two different complex conjugates, hence imaginary and unequal.