For what value(s) of b will x^2 + bx + 3 have exactly one root?

Question

I'm rather new at this but wouldn't b=2*sqrt(3) give (x+sqrt3)² as factors and that would give one root?
Check my thinking.

for a quadratic equation to have exactly one root, the discriminat must be zero i.e

if equation is:
ax^2+bx+c=0 , then for exactly one root,
b^2-4ac=0
i.e.
b^2=4ac
for your equation,
x^2+bx+3=0
b^2=4(1)(3)
=12
hence
b=sqrt(12)
=2(sqrt(3))