For univariate Gaussian binary classification: Optimal decision regions are defined by decision boundaries in $x$ where $P(c_1)P(x|\theta_1) = P(c_2)P(x|\theta_2)$.
<latex> \begin{align*}
P(c_1)\frac{1}{\sqrt{2\pi}\sigma_1} e^{-\frac{1}{2\sigma_1^2}(\mu_1-x)^2} &=
\end{align*} </latex>
Cancel common terms and take the log
<latex>
\begin{align*}
P(c_1)\sigma_2^2 e^{-\frac{1}{2\sigma_1^2}(\mu_1-x)^2} &=
\log P(c_1) + \log \sigma_2 - \frac{1}{2\sigma_1^2}(\mu_1-x)^2 &=
\frac{1}{2\sigma_2^2}(\mu_2-x)^2 - \frac{1}{2\sigma_1^2}(\mu_1-x)^2&=
\frac{1}{\sigma_2^2}(\mu_2-x)^2 - \frac{1}{\sigma_1^2}(\mu_1-x)^2 &=
\sigma_1^2(\mu_2-x)^2 - \sigma_2^2(\mu_1-x)^2 &=
\sigma_1^2(\mu_2^2 - 2\mu_2x + x^2) - \sigma_2^2(\mu_1^2 - 2x\mu_1 + x^2) &=
\sigma_1^2\mu_2^2 - 2\sigma_1^2\mu_2x + \sigma_1^2x^2 - \sigma_2^2\mu_1^2 + 2\sigma_2^2\mu_1x - \sigma_2^2x^2&=
\sigma_1^2x^2 - \sigma_2^2x^2 + 2\sigma_2^2\mu_1x - 2\sigma_1^2\mu_2x - \sigma_2^2\mu_1^2 + \sigma_1^2\mu_2^2 &=
\end{align*} </latex>
Use the quadratic formula:
<latex> \begin{align*}
x &= \frac{-b \pm \sqrt(b^2-4ac)}{2a}
\end{align*}
\begin{align*}
b &= 2(\sigma_2^2\mu_1 - \sigma_1^2\mu_2)
c &= \sigma_1^2\mu_2^2 - \sigma_2^2\mu_1^2 - 2 \sigma_1^2 \sigma_2^2 \log \frac{P(c_2) \sigma_1 }{ P(c_1) \sigma_2 }
\end{align*}
</latex>