Question

Find the values of k for which the given quadratic equation has real and distinct roots:

(i) kx² + 2x + 1 = 0

(ii) kx² + 6x + 1 = 0

(iii) x² - kx + 9 = 0

Answer 1

(i) kx² + 2x + 1 = 0

For a quadratic equation, ax² + bx + c = 0,

D = b² – 4ac 

If D > 0, roots are real and distinct

kx² + 2x + 1 = 0

⇒ D = 4 – 4k 

⇒ 4 – 4k > 0 

⇒ k < 1

(ii) kx² + 6x + 1 = 0

For a quadratic equation, ax² + bx + c = 0, 

D = b² – 4ac 

If D > 0, roots are real and distinct

kx² + 6x + 1 = 0

⇒ D = 36 – 4k 

⇒ 36 – 4k > 0 

⇒ k < 9

(iii) x² - kx + 9 = 0

For a quadratic equation, ax² + bx + c = 0,

D = b² – 4ac

If D > 0, roots are real and distinct

x² - kx + 9 = 0

⇒ D = k² – 36 

⇒ k² – 36 > 0 

⇒ (k + 6)(k – 6) > 0 

⇒ k < -6 or k > 6