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\begin{document}

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\section*{quadratic formula}

\hypertarget{the_quadratic_formula}{}\section*{{The quadratic formula}}\label{the_quadratic_formula}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{discussion}{Discussion}\dotfill \pageref*{discussion} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Consider the equation

\begin{equation}
a{x}^2 + b{x} + c = 0 ,
\label{eqn}\end{equation}
which we wish to solve for $x$. In certain contexts, the solutions are given by one or more versions of the quadratic formula.

\hypertarget{discussion}{}\subsection*{{Discussion}}\label{discussion}

The coefficients $a, b, c$ are commonly taken from an [[algebraically closed field]] $K$ of [[characteristic]] $0$, such as the field $\mathbb{C}$ of [[complex numbers]], although any quadratically closed field whose characteristic is not $2$ would work just as well. Alternatively, the coefficients can be taken from a [[real closed field]] $K$, such as the field $\mathbb{R}$ of [[real numbers]]; then the solutions belong to $K[\mathrm{i}]$. (Of course, $\mathbb{R}[\mathrm{i}]$ is simply $\mathbb{C}$ again.) More generally, starting from any [[integral domain]] $K$ whose characteristic is not $2$, the solutions belong to some [[splitting field]] of $K$. (Of course, there are solutions in \emph{some} splitting field, regardless of the characteristic, but they are not given by the quadratic formula if the characteristic is $2$.)

Explicitly, the solutions of \eqref{eqn} may be given by the \textbf{usual quadratic formula}

\begin{equation}
x_\pm = \frac{-b \pm \sqrt{b^2 - 4a{c}}}{2a} ,
\label{usual}\end{equation}
which works as long as $a \ne 0$. There is also an \textbf{alternate quadratic formula}

\begin{equation}
x_\pm = \frac{2c}{-b \mp \sqrt{b^2 - 4a{c}}} ,
\label{alt}\end{equation}
which may be obtained from \eqref{usual} by rationalizing the numerator; this works as long as $c \ne 0$. (Note that $\pm$ and $\mp$ appear here simply to indicate the two [[square roots]] of the determinant $b^2 - 4a{c}$ and how they correspond to the two solutions $x_\pm$; we do not need to have a [[function]] $\sqrt{}$ which always chooses a `principal' square root.)

These two formulas are reconciled in the [[projective line]] of $K$. As long as $(a, b, c) \ne (0, 0, 0)$, there are two solutions (which might happen to be equal) in the projective line. If $a = 0$, then one of these solutions is $\infty$, and \eqref{usual} correctly gives us that solution (as long as $b \ne 0$) for one choice of square root, although it gives $0/0$ for the other choice. Similarly, \eqref{alt} correctly gives us $x = 0$ when $c = 0$ and $b \ne 0$, but it does not give us the other root when $c = 0$. Note that if $a, c = 0$ but $b \ne 0$, then \eqref{usual} gives us one root ($\infty$) while \eqref{alt} gives us the other ($0$).

So in general, we should be given $a \ne 0$, $b \ne 0$, or $c \ne 0$ for a nondegenerate equation \eqref{eqn}. If $a \ne 0$, then we use \eqref{usual}; if $c \ne 0$, then we use \eqref{alt}. Finally, if $b \ne 0$, then we use both; each root will be successfully given by at least one formula for some choice of square root of $b^2 - 4a{c}$.

When the coefficients come from an [[ordered field]] $K$ (which we assume real closed), then we can write down a formula specially for the case when $b \ne 0$. This is the \textbf{numerical analysts' quadratic formula}

\begin{equation}
\begin {gathered}
   \displaystyle x_{\hat{b}} = \frac{2c}{-b - \hat{b}\sqrt{b^2 - 4a{c}}} ;\\
   \displaystyle x_{-\hat{b}} = \frac{-b - \hat{b}\sqrt{b^2 - 4a{c}}}{2a} .\\
\end {gathered}
\label{numanal}\end{equation}
In this formula, $\hat{b}$ is the sign of $b$, that is $b/{|b|}$; also, we must choose a nonnegative principal square root, so that $\sqrt{b^2 - 4a{c}} \lt 0$ in $K$ is avoided (and thus the common denominator of $x_{\hat{b}}$ and numerator of $x_{-\hat{b}}$ is nonzero even if not imaginary). Despite the name, this formula is not sufficient for all purposes in [[numerical analysis]]; one still needs all three formulas and chooses between them based on whether $a \ne 0$, $b \ne 0$, or $c \ne 0$ is best established.

There is also an interesting issue about whether $b^2 - 4a{c} \ne 0$. Everything above is valid in weak forms of [[constructive mathematics]], except for the statement that $\mathbb{C}$ is algebraically closed. That claim follows from [[weak countable choice]] ($WCC$), which in turn will follow from either [[excluded middle]] or [[countable choice]], which is accepted by most constructive mathematicians. Nevertheless, the statement

\begin{displaymath}
\forall\, a, b, c\colon \mathbb{C},\; \exists\, r\colon \mathbb{C},\; r^2 = b^2 - 4a{c}
\end{displaymath}
is false in (for example) the [[internal language]] of the [[sheaf topos]] over the [[real line]]. (Essentially, this is because there is no [[continuous map]] $\sqrt{}$ on any neighbourhood of $0$ in $\mathbb{C}$.) If we are given that $a, b, c$ are real, or if we are given that $b^2 \ne 4a{c}$, then there is no problem. But in general, we cannot define this square root, which appears in every version of the quadratic formula.

However, there is a more subtle sense in which $\mathbb{C}$ is algebraically closed even without $WCC$; essentially, this allows us to approximate the [[subset]] of $\mathbb{C}$ whose elements are the two solutions of \eqref{eqn} (using two-element subsets of the field of, say, [[Gaussian numbers]]) even if we can't approximate any one solution (using individual, say, Gaussian numbers); see \hyperlink{Richman}{Richman (1998)} for details. The quadratic formula can then be interpreted as indicating this approximated subset.

Sometimes one considers the equation

\begin{displaymath}
a{x}^2 + 2p{x} + c = 0
\end{displaymath}
instead of \eqref{eqn}; then \eqref{usual} simplifies to

\begin{equation}
x_\pm = \frac{-p \pm \sqrt{p^2 - a{c}}}a
\label{simpl}\end{equation}
(and similarly for \eqref{alt} and \eqref{numanal}). This is valid even in characteristic $2$, but unfortunately then it is fairly useless, since $b = 2p = 0$. More precisely, if $b = 0$, then \eqref{simpl} with $p = 0$ gives the roots $\pm\sqrt{-c/a}$ in any characteristic, but in that case the equation was easy to solve without any formula. On the other hand, if $b \ne 0$ and $\char K = 2$, then no version of the quadratic formula is applicable, yet this gives no information as to whether the polynomial is [[solvable polynomial|solvable]] and what its roots are if it is. For example, the equation $x^2 + x = 0$ has roots $0$ and $1$ in $\F_2$ (or $0$ and $-1$ in any field, as may be found by factoring), while $x^2 + x + 1 = 0$ is not solvable over $\F_2$, yet both have $b^2 - 4a{c} = 1$ and give $0/0$ in both \eqref{usual} and \eqref{alt} (while \eqref{numanal} and \eqref{simpl} are directly inapplicable).

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Fred Richman]]; 1998; \emph{The fundamental theorem of algebra: a constructive development without choice}; \href{http://math.fau.edu/richman/html/docs.htm}{Fred Richman's Documents}

\end{itemize}
[[!redirects quadratic formula]] [[!redirects quadratic formulas]] [[!redirects quadratic formulae]] [[!redirects quadratic formulæ]]



\end{document}