nLab quadratic formula

The quadratic formula

The quadratic formula

Idea

Consider the quadratic function

(1)f(x)ax 2+bx+c f(x) \coloneqq a{x}^2 + b{x} + c

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.

Discussion

The coefficients a,b,ca, b, c of the quadratic function are commonly taken from an algebraically closed field KK of characteristic 00, such as the field \mathbb{C} of complex numbers, although any quadratically closed field whose characteristic is not 22 would work just as well. Alternatively, the coefficients can be taken from a real closed field KK, such as the field \mathbb{R} of real numbers; then the solutions belong to K[i]K[\mathrm{i}]. (Of course, [i]\mathbb{R}[\mathrm{i}] is simply \mathbb{C} again.) More generally, starting from any integral domain KK whose characteristic is not 22, the solutions belong to some splitting field of KK. (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 22.)

Forms of the formula

Explicitly, the zero set of (1) may be given by the usual quadratic formula

(2)x ±=b±b 24ac2a, x_\pm = \frac{-b \pm \sqrt{b^2 - 4a{c}}}{2a} ,

which works as long as a0a \ne 0. There is also an alternate quadratic formula

(3)x ±=2cbb 24ac, x_\pm = \frac{2c}{-b \mp \sqrt{b^2 - 4a{c}}} ,

which may be obtained from (2) by rationalizing the numerator; this works as long as c0c \ne 0. (Note that ±\pm and \mp appear here simply to indicate the two square roots of the determinant b 24acb^2 - 4a{c} and how they correspond to the two solutions x ±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 KK. As long as (a,b,c)(0,0,0)(a, b, c) \ne (0, 0, 0), there are two solutions (which might happen to be equal) in the projective line. If a=0a = 0, then one of these solutions is \infty, and (2) correctly gives us that solution (as long as b0b \ne 0) for one choice of square root, although it gives 0/00/0 for the other choice. Similarly, (3) correctly gives us x=0x = 0 when c=0c = 0 and b0b \ne 0, but it does not give us the other root when c=0c = 0. Note that if a,c=0a, c = 0 but b0b \ne 0, then (2) gives us one root (\infty) while (3) gives us the other (00).

So in general, we should be given a0a \ne 0, b0b \ne 0, or c0c \ne 0 for a nondegenerate equation (1). If a0a \ne 0, then we use (2); if c0c \ne 0, then we use (3). Finally, if b0b \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 24acb^2 - 4a{c}.

When the coefficients come from an ordered field KK (which we assume real closed), then we can write down a formula specially for the case when b0b \ne 0. This is the numerical analysts' quadratic formula

(4)x b^=2cbb^b 24ac; x b^=bb^b 24ac2a. \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}

In this formula, b^\hat{b} is the sign of bb, that is b/|b|b/{|b|}; also, we must choose a nonnegative principal square root, so that b 24ac<0\sqrt{b^2 - 4a{c}} \lt 0 in KK is avoided (and thus the common denominator of x b^x_{\hat{b}} and numerator of x b^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 a0a \ne 0, b0b \ne 0, or c0c \ne 0 is best established.

Constructive issues

Everything above is valid in weak forms of constructive mathematics if one takes C\mathbb{C} \coloneqq \mathbb{C}_C to be the modulated Cauchy complex numbers, see Ruitenberg (1991).

However, if one takes D\mathbb{C} \coloneqq \mathbb{C}_D to be the Dedekind complex numbers, there is an interesting issue about whether b 24ac0b^2 - 4a{c} \ne 0. Everything above is valid in weak forms of constructive mathematics except for the statement that D\mathbb{C}_D is quadratically closed?. That claim follows from weak countable choice (WCCWCC), which in turn will follow from either excluded middle or countable choice. Nevertheless, the statement

a,b,c: D,r: D,r 2=b 24ac \forall\, a, b, c\colon \mathbb{C}_D,\; \exists\, r\colon \mathbb{C}_D,\; r^2 = b^2 - 4a{c}

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 00 in D\mathbb{C}_D.) If we are given that a,b,ca, b, c are real, or if we are given that b 24acb^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 D\mathbb{C}_D is algebraically closed even without WCCWCC; essentially, this allows us to approximate the subset of D\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.

Simplified formulas and characteristic 22

Sometimes one considers the quadratic function

f(x)ax 2+2px+cf(x) \coloneqq a{x}^2 + 2p{x} + c

instead of (1); then (2) simplifies to

(5)x ±=p±p 2aca x_\pm = \frac{-p \pm \sqrt{p^2 - a{c}}}a

(and similarly for (3) and (4)).

This is valid even in characteristic 22, but unfortunately then it is fairly useless, since b=2p=0b = 2p = 0. More precisely, if b=0b = 0, then (5) with p=0p = 0 gives the roots ±c/a\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 b0b \ne 0 and charK=2\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)x 2+xf(x) \coloneqq x^2 + x has roots 00 and 11 in F 2\F_2 (or 00 and 1-1 in any field, as may be found by factoring), while f(x)x 2+x+1f(x) \coloneqq x^2 + x + 1 is not solvable over F 2\F_2, yet both have b 24ac=1b^2 - 4a{c} = 1 and give 0/00/0 in both (2) and (3) (while (4) and (5) are directly inapplicable).

See also

References

  • 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 16:04:48. See the history of this page for a list of all contributions to it.