Question

If k is real, discuss the nature of the roots of the polynomial equation 2x² + kx + k = 0, in terms of k.

Answer 1

The given quadratic equation is 2x² + kx + k = 0 

a = 2, b = k, c = k 

∆ = b² – 4ac = k² – 4(2) k = k² – 8k 

(i) If the roots are equal k² – 8k = 0 

⇒ k(k – 8) = 0 

⇒ k = 0, k = 8 

(ii) If the roots are real

k² – 8k > 0 

k(k – 8) > 0 

k ∈ (-∞, 0) ∪ (8, ∞)

(iii) If this roots are imaginary 

k² – 8k < 0 

⇒ k ∈ (0, 8)