Answer
The solutions are $-\displaystyle \frac{3}{2}+i\frac{\sqrt{47}}{2}$ and $-\displaystyle \frac{3}{2}-i\frac{\sqrt{47}}{2}$.
Work Step by Step
$ x^{2}+3x+14=0\qquad$ ...Write left side in the form $x^{2}+bx.$(add $-14$ to each side)
$ x^{2}+3x=-14\qquad$ ...square half the coefficient of $x$.
$(\displaystyle \frac{3}{2})^{2}=\frac{9}{4}\qquad$ ...complete the square by adding$ \displaystyle \frac{9}{4}$ to each side of the expression
$ x^{2}+3x+\displaystyle \frac{9}{4}=-14+\frac{9}{4}\qquad$ ...Write left side as a binomial squared.
$(x+\displaystyle \frac{3}{2})^{2}=\frac{-47}{4}\qquad$ ...take square roots of each side.
$ x+\displaystyle \frac{3}{2}=\pm i\sqrt{\frac{47}{4}}\qquad$ ...simplify $i\displaystyle \sqrt{\frac{47}{4}}=i\frac{\sqrt{47}}{\sqrt{4}}=i\frac{\sqrt{47}}{2}$
$ x+\displaystyle \frac{3}{2}=\pm i\frac{\sqrt{47}}{2}\qquad$ ...add $-\displaystyle \frac{3}{2}$ to each side.
$x=-\displaystyle \frac{3}{2}\pm i\frac{\sqrt{47}}{2}$
