Question

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

3x² – 2x + 5 = 0

Answer 1

Given: 3x² – 2x + 5 = 0

Step-wise calculation:

1. Identify coefficients:

a = 3, b = –2, c = 5

2. Compute discriminant:

Δ = b² – 4ac 

= (–2)² – 4 × 3 × 5 

= 4 – 60

= –56

3. The discriminant test states: if Δ < 0 the roots are imaginary (non‑real); if Δ = 0 the roots are real and equal; if Δ > 0 the roots are real and unequal.

4. Here Δ < 0 and Δ ≠ 0, so `sqrt(Δ) = i xx sqrt(56)` is nonzero; the quadratic formula `x = (-b ± sqrt(Δ))/(2a)` gives two values with different signs on the imaginary part, hence two distinct (unequal) complex numbers which are conjugates of each other.

Because the discriminant is negative and not zero, the equation has two unequal imaginary (non‑real complex conjugate) roots.