If 2 + i.sqrt(3) is a root of x^2 + px + q = 0 (both p and q are real), then find the values of p and q.

Given that [2 + i.sqrt(3)] is a root of x^2 + px + q = 0

We know that if a quadratic equation has non-real (imaginary) roots, its roots always exist in conjugate pairs. for eg., (a + ib) and (a - ib)

Also note that i = -sqrt(1) and because it doesn't really exists or it can't be located on the real number line, its said to be a non real or an imaginary or a complex number. This forms a completely new branch called Complex Numbers, whose questions and basics will be discussed later.

According to the rule of conjugate pairs, the roots of x^2 + px + q = 0 are 

[2 + i.sqrt(3)]  and [2 - i.sqrt(3)]

Now, for x^2 + px + q = 0;

sum or roots = -p

=> [2 + i.sqrt(3)] + [2 - i.sqrt(3)] = -p

=> 4 = -p

=> p = -4

Similarly, 

Product of roots = q

=> [2 + i.sqrt(3)]*[2 - i.sqrt(3)] = q

=> (2)^2 - (i.sqrt(3))^2 = q               (using (a+b)(a-b) = a^2 - b^2)

=> 4 - i^2 3= q

=> 4 - (-3) = q

=> q = 7

So, p= -4 and q = 7