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 ∈ ${\displaystyle (-\infty ,0)\cup (8,\infty )}$

(iii) If this roots are imaginary

k² – 8k < 0

⇒ k ∈ (0, 8)