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\Huge\sf \textbf{Solving the general quadratic equation}\\
\textcolor{red}{Where does the quadratic formula come from?}
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{\sf
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We want to find \textcolor{blue}{all values $x$} such that $ax^2+bx+c=0$.

If $a=0$ then we can easily solve $bx+c=0$\newline
(\textcolor{blue}{unless} $b=0\ne c$, in which case there are no solutions).

So \textcolor{blue}{assume $a\ne 0$}, and divide by $a$ on both sides.

$\Leftrightarrow \ x^2+\frac bax+\frac ca=0$.\quad Now \textcolor{green}{complete the square}

$\Leftrightarrow \ x^2+2\frac b{2a}x+\textcolor{green}{\boxed{\phantom{\frac{b^2}{4a^2}}}}-\textcolor{green}{\boxed{\phantom{\frac{b^2}{4a^2}}}}+\frac ca=0$

$\Leftrightarrow \ (x+\frac b{2a})^2=\textcolor{green}{\boxed{\phantom{\frac{b^2}{4a^2}}}}-\frac{c}{a}=\frac{\textcolor{green}{\boxed{\phantom{b^2}}}-4ac}{4a^2}$

$\Leftrightarrow \ x+\frac b{2a}=\pm\sqrt{\frac{\textcolor{green}{\boxed{\phantom{b^2}}}-4ac}{4a^2}}=
\frac{\pm\sqrt{b^2-4ac}}{2a}$
}
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{\huge$x=\frac{-b+\sqrt{b^2-4ac}}{2a}$}

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{\huge$x=\frac{-b-\sqrt{b^2-4ac}}{2a}$}
}
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\begin{center}\tiny 
Fill in the blanks --- Math Poster 2015 --- math.chapman.edu
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}
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