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\documentclass{ltx-talk}

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\title{The Quadratic Formula}
\subtitle{$ax^{2} + bx + c = 0$}
\author{David}
\institute{The \LaTeX{} Project}
\date{2025-12-06}

\begin{document}

\maketitle


\begin{frame}
  \frametitle{Completing the Square}

  \large

  First, complete the square
  
  \begin{itemize}[action-spec = <+->, item-vspace = 20pt, begin-vspace = 20pt]
    \item $ax^{2} + bx +c=0$
    \item $x^{2} + \frac{b}{a}x + \frac{c}{a} = 0$
    \item $x^{2} + \frac{b}{a}x + \bigl(\frac{b}{2a}\bigr)^{2} 
      = -\frac{c}{a} + \bigl(\frac{b}{2a}\bigr)^{2}$
    \item $\bigl(x + \frac{b}{2a}\bigr)^{2} 
      = -\frac{c}{a} + \bigl(\frac{b}{2a}\bigr)^{2}$
  \end{itemize}
\end{frame}

\begin{frame}
 \frametitle{Solve for~$x$}
 \large

 Now, solve for~$x$
 
  \begin{itemize}[action-spec = <+->, item-vspace = 20pt, begin-vspace = 20pt]
    \item $\bigl(x + \frac{b}{2a}\bigr)^{2}
      = -\frac{c}{a} + \bigl(\frac{b}{2a}\bigr)^{2}$
    \item $x + \bigl(\frac{b}{2a}\bigr)
      = \pm \sqrt{-\frac{c}{a} + \bigl(\frac{b}{2a}\bigr)^{2}}$
    \item $x = -\bigl(\frac{b}{2a}\bigr)
      \pm \sqrt{-\frac{c}{a} + \bigl(\frac{b}{2a}\bigr)^{2}}$
    \item $\displaystyle x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$
  \end{itemize}
    
\end{frame}

\begin{frame}
 \frametitle{The Quadratic Formula}
 \large

 The solution of the quadratic equation.

 \[
   ax^{2} + bx + c = 0
 \]

 where~$a$ is non-zero, is given by the formula

  \[
    x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}
  \]

\end{frame}

\end{document}
