## Precalculus (6th Edition) Blitzer

The solution of the equation $2{{x}^{2}}+2x+3=0$ in standard form is $\left\{ -\frac{1}{2}+\frac{\sqrt{5}}{2}i,-\frac{1}{2}-\frac{\sqrt{5}}{2}i \right\}$.
Compare the equation $2{{x}^{2}}+2x+3=0$ with $a{{x}^{2}}+bx+c$. \begin{align} & a=2 \\ & b=2 \\ & c=3 \end{align} Substitute $a=2$, $b=2$ and $c=3$ in the formula $x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$. \begin{align} & x=\frac{-2\pm \sqrt{{{2}^{2}}-4\left( 2 \right)\left( 3 \right)}}{2\left( 2 \right)} \\ & =\frac{-2\pm \sqrt{4-24}}{4} \\ & =\frac{-2\pm \sqrt{-20}}{4} \end{align} Use the property $\sqrt{-b}=i\sqrt{b}$. \begin{align} & x=\frac{-2\pm i\sqrt{20}}{4} \\ & =\frac{-2\pm i\sqrt{4\cdot 5}}{4} \\ & =\frac{-2\pm 2i\sqrt{5}}{4} \end{align} Express the complex number in the standard form. \begin{align} & x=-\frac{2}{4}\pm \frac{2\sqrt{5}}{4}i \\ & =-\frac{1}{2}\pm \frac{\sqrt{5}}{2}i \end{align} Therefore, the solution of the equation $2{{x}^{2}}+2x+3=0$ in standard form is $\left\{ -\frac{1}{2}+\frac{\sqrt{5}}{2}i,-\frac{1}{2}-\frac{\sqrt{5}}{2}i \right\}$.