If 3 is a root of the quadratic equation x2 – x + k = 0, find the value of p so that the roots of the equation x2 + k (2x + k + 2) + p = 0 are equal.
Given 3 is a root of the quadratic equation x2 – x + k = 0
(3)2 – 3 + k = 0
k + 6 = 0
k = – 6
Given equation is x2 + k (2x + k + 2) + p = 0
x2 + 2kx + (k2 + 2k + p) = 0
Comparing with standard quadratic equation ax2 + bx + c = 0
a = 1 b = 2k c = k2 + 2k + p
Given that the roots of equation are equal
Thus D = 0
Discriminant D = b2 – 4ac = 0
(2k)2 – 4.1. (k2 + 2k + p) = 0
4k2 – 4k2 – 8k – 4p = 0
– 8k – 4p = 0
4p = – 8k
p = – 2k
p = – 2. – 6 = 12
p = 12
The value of p is – 12 for which roots of the quadratic equation are equal.