Question

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

5x² – 13x – 6 = 0

Answer 1

Given: 5x² – 13x – 6 = 0

Step-wise calculation:

1. Identify coefficients:

a = 5, b = –13, c = –6

2. Discriminant D = b² – 4ac

= (–13)² – 4 × 5 × (–6) 

= 169 + 120

= 289

3. By the discriminant test, D > 0 implies two distinct real roots D = 0 would give equal roots; D < 0 gives nonreal roots.

4. Here, D = 289 = 17², so `sqrt(D) = 17` is an integer. 

The quadratic formula gives roots `x = (-b ± sqrt(D))/(2a) = (13 ± 17)/10`, i.e. `x = 30/10 = 3` and `x = (-4)/10 = (-2)/5`. Because each root is a ratio of integers, both roots are rational.

Because the discriminant is positive so the roots are real and unequal and it is a perfect square so `sqrt(D)` is an integer, making the roots rational.