Consider the quadratic function
which we wish to find the elements of the zero set. In certain contexts, the elements of the zero set are given by one or more versions of the quadratic formula.
The coefficients $a, b, c$ of the quadratic function 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 some splitting field, regardless of the characteristic, but they are not given by the quadratic formula if the characteristic is $2$.)
Explicitly, the zero set of (1) may be given by the usual quadratic formula
which works as long as $a \ne 0$. There is also an alternate quadratic formula
which may be obtained from (2) 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 (2) 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, (3) 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 (2) gives us one root ($\infty$) while (3) 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 (1). If $a \ne 0$, then we use (2); if $c \ne 0$, then we use (3). 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 numerical analysts' quadratic formula
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.
Everything above is valid in weak forms of constructive mathematics if one takes $\mathbb{C} \coloneqq \mathbb{C}_C$ to be the modulated Cauchy complex numbers, see Ruitenberg (1991).
However, if one takes $\mathbb{C} \coloneqq \mathbb{C}_D$ to be the Dedekind complex numbers, there is 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 $\mathbb{C}_D$ is quadratically closed?. That claim follows from weak countable choice ($WCC$), which in turn will follow from either excluded middle or countable choice. Nevertheless, the statement
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 $\mathbb{C}_D$.) 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 $\mathbb{C}_D$ is algebraically closed even without $WCC$; essentially, this allows us to approximate the subset of $\mathbb{C}_D$ whose elements are the two solutions of (1) (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 Richman (1998) for details. The quadratic formula can then be interpreted as indicating this approximated subset.
Sometimes one considers the quadratic function
instead of (1); then (2) simplifies to
(and similarly for (3) and (4)).
This is valid even in characteristic $2$, but unfortunately then it is fairly useless, since $b = 2p = 0$. More precisely, if $b = 0$, then (5) 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 and what its roots are if it is. For example, the quadratic function $f(x) \coloneqq x^2 + x$ has roots $0$ and $1$ in $\F_2$ (or $0$ and $-1$ in any field, as may be found by factoring), while $f(x) \coloneqq x^2 + x + 1$ is not solvable over $\F_2$, yet both have $b^2 - 4a{c} = 1$ and give $0/0$ in both (2) and (3) (while (4) and (5) are directly inapplicable).
Wim Ruitenberg, Constructing Roots of Polynomials over the Complex Numbers, Computational Aspects of Lie Group Representations and Related Topics, CWI Tract, Vol. 84, Centre for Mathematics and Computer Science, Amsterdam, 1991, pp. 107–128. (pdf)
Fred Richman; 1998; The fundamental theorem of algebra: a constructive development without choice; Fred Richman’s Documents
Last revised on June 2, 2022 at 20:04:48. See the history of this page for a list of all contributions to it.