Question

Find the nature of the roots of the quadratic equation x² – 5x + 9 = 0.

Answer 1

Given: x² – 5x + 9 = 0

Step-wise calculation:

1. Identify coefficients:

a = 1, b = –5, c = 9

2. Compute the discriminant

D = b² – 4ac 

= (–5)² – 4(1)(9) 

= 25 – 36

= –11

3. Since D < 0, the quadratic has no real roots and instead two distinct complex (conjugate) roots (criterion: D < 0 ⇒ nonreal complex roots).

4. By the quadratic formula `x = (-b ± sqrt(D))/(2a)`

The roots are `x = (5 ± sqrt(-11))/2` 

= `5/2 ± (isqrt(11))/2`

The equation has two distinct nonreal complex conjugate roots: `x = 5/2 ± (isqrt(11))/2`.