Answer
$z = (-3+\sqrt 5)/2$
Work Step by Step
$z^2+3z+1=0$
$z^2+3z+1+(3/2)^2=(3/2)^2$
$z^2+3z+1+9/4=9/4/$
$z^2+3z+9/4 +1=9/4$
$(z+3/2)^2+1=9/4$
$(z+3/2)^2+1-1=9/4-1$
$(z+3/2)^2 = 5/4$
$\sqrt{(z+3/2)^2} =\sqrt{5/4}$
$z+3/2 = \sqrt {5/4}$
$z+3/2-3/2 = \sqrt {5/4}-3/2$
$z= -3/2 +\sqrt {5/4}$
$z= -3/2 +\sqrt 5/\sqrt 4$
$z= -3/2 +\sqrt 5/2$
$z = (-3+\sqrt 5)/2$