$\begin{array}{l}
{x^2} + 3x - 5 = 0\\
\Leftrightarrow {x^2} + 2.x.\frac{3}{2} + \frac{9}{4} - 5 - \frac{9}{4}\\
\Leftrightarrow {\left( {x + \frac{3}{2}} \right)^2} = \frac{{29}}{4}\\
\Leftrightarrow x + \frac{3}{2} = \frac{{ + - \sqrt {29} }}{2}\\
TH1:x + \frac{3}{2} = \frac{{\sqrt {29} }}{2}\\
x = \frac{{\sqrt {29} - 3}}{2}\\
TH2:x + \frac{3}{2} = \frac{{ - \sqrt {29} }}{2}\\
x = \frac{{ - \sqrt {29} - 3}}{2}
\end{array}$