Question
Let $$p,q \in R.\,\,{\text{If }}\,{\text{2}} - \sqrt 3 $$ is a root of the quadratic equation, $${x^2} + px + q = 0,$$ then;
A.
$${p^2} - 4q + 12 = 0$$
B.
$${q^2} - 4p - 16 = 0$$
C.
$${q^2} + 4p + 14 = 0$$
D.
$${p^2} - 4q - 12 = 0$$
Answer :
$${p^2} - 4q - 12 = 0$$
Solution :
Since $$\,2 - \sqrt 3 $$ is a root of the quadratic equation
$${x^2} + px + q = 0$$
$$\therefore \,\,2 + \sqrt 3 \,$$ is the root of unity
⇒ Sum of roots = 4, Product of roots = 1
$$\eqalign{
& \Rightarrow \,p = - 4,q = 1 \cr
& \Rightarrow \,{p^2} - 4q - 12 \cr
& = 16 - 4 - 12 \cr
& = 0 \cr} $$