Question

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

2x² – 5x – 4 = 0

Answer 1

Given: 2x² – 5x – 4 = 0 (a = 2, b = –5, c = –4)

Step-wise calculation:

1. Compute the discriminant:

D = b² – 4ac 

= (–5)² – 4 × 2 × (–4) 

= 25 + 32

= 57

D > 0, so the quadratic has two distinct (unequal) real roots.

2. Check whether the square root of the discriminant is rational:

57 is not a perfect square (49 < 57 < 64), so `sqrt(57)` is irrational. 

For rational coefficients a, b, c, when D > 0 but D is not a perfect square, the two roots `(-b ± sqrt(D))/(2a)` are irrational conjugates.

Because D = 57 > 0 the roots are real and unequal; because `sqrt(57)` is irrational the roots are irrational.

Hence, irrational and unequal is correct.