Find the values of k for which the given quadratic equation has real and distinct roots:
(i) Given: kx2 + 6x + 1 = 0
Comparing with standard quadratic equation ax2 + bx + c = 0
a = k b = 6 c = 1
For real and distinct roots: D > 0
Discriminant D = b2 – 4ac > 0
62 – 4k > 0
36 – 4k > 0
4k < 36
k < 9
(ii) Given: x2 – kx + 9 = 0
Comparing with standard quadratic equation ax2 + bx + c = 0
a = 1 b = – k c = 9
For real and distinct roots: D > 0
Discriminant D = b2 – 4ac > 0
(– k)2 – 4.1.9 = k2 – 36 > 0
k2 > 36
k > 6or k < – 6 taking square root both sides
(iii) 9x2 + 3kx + 4 = 0
Comparing with standard quadratic equation ax2 + bx + c = 0
a = 9 b = 3k c = 4
For real and distinct roots: D > 0
Discriminant D = b2 – 4ac > 0
(3k)2 – 4.4.9 = 9k2 – 144 > 0
9k2 > 144
k2 > 16
k > 4ork < – 4 taking square root both sides
(iv) 5x2 – kx + 1 = 0
Comparing with standard quadratic equation ax2 + bx + c = 0
a = 5 b = – k c = 1
For real and distinct roots: D > 0
Discriminant D = b2 – 4ac > 0
(– k)2 – 4.5.1 = k2 – 20 > 0
k2 > 20
k > 2√5 or k < –2√5 taking square root both sides