Where two roots run into each other

Completing the square solves one quadratic at a time. It takes $x^2 + px + q$, finds the missing corner, and reads off a side. What it never does is look at all the quadratics at once, and that is a different kind of question with a different kind of answer.

A monic quadratic is fixed by two numbers, $p$ and $q$. So the whole family of them is a plane: every point $(p, q)$ is one equation, and every equation is one point. Asking which equations have real roots is asking for a region of that plane, and the answer turns out to be a region with a boundary that can be drawn — a parabola — together with a reason for the parabola that can also be drawn, as a family of straight lines that all touch it.

Three curves against one axis #

Before the plane, the familiar picture, because the plane is a way of reorganising it.

Hold $p$ fixed and move $q$. The graph of $y = x^2 + px + q$ keeps its shape and its axis of symmetry and simply slides up the page by whatever $q$ adds. Low down it crosses the $x$-axis twice; at one particular height it rests on the axis at a single point; above that it clears the axis entirely.

The middle curve is the interesting one. It is the moment the two crossing points, which were approaching one another as the curve rose, meet at the bottom of the bowl and vanish together. The two roots do not disappear one at a time; they collide and annihilate in pairs. That is the whole reason the count of real roots of a quadratic jumps from two to nought with nothing in between except the single instant where it is one.

The quantity that detects the collision is the discriminant, $p^2 - 4q$. In the square-completing picture it is the area of the completed square: positive when there is a square whose side can be taken, nought when the square has shrunk to nothing, negative when the construction would need a negative area. What the coefficients already know reads it another way, as the squared difference of the roots, $(r - s)^2$, which is zero precisely when the two roots coincide. Both readings say the same thing: the discriminant measures how far apart the roots are, and its vanishing is their meeting.

A plane in which each equation is a point #

Now stop sliding one curve and look at every curve at once. The point $(p, q)$ is the equation; the question “does it have real roots” is a yes-or-no label attached to each point; and the labels are decided by one inequality, $q \le p^2/4$.

That is the figure at the top. The boundary between the two answers is the parabola $q = p^2/4$, and it is worth being clear what kind of object it is, because it is easy to confuse with the parabolas in the previous figure. Those were graphs of individual equations, living in a plane whose coordinates are $x$ and $y$. This one lives in the plane of coefficients, and each of its points is a whole equation — specifically, an equation with a repeated root. The equation $x^2 - 2x + 1$ is the point $(-2, 1)$, and it lies exactly on the curve, because it is $(x - 1)^2$.

So the parabola is the set of perfect squares. Every point on it is some $(x - t)^2 = x^2 - 2tx + t^2$, which is the point $(-2t, t^2)$, and as $t$ runs over the real numbers that point traces $q = p^2/4$. The curve is literally the list of equations that completing the square finishes with nothing left over.

Below the curve lie equations whose completed square has room to spare, above it equations whose completed square would need to be negative. The picture turns “the discriminant is positive” from an arithmetic condition into a place, and places can be looked at: the region with real roots contains the whole lower half-plane, and the region without them is the inside of the bowl, which starts at the origin and widens as it rises. An equation with $q < 0$ always has real roots — a parabola that starts below the axis at $x = 0$ must cross it — and that fact is visible as the lower half-plane lying entirely inside the shaded region.

Every root is a straight line #

Here is the surprise, and it is the reason to draw the plane at all.

Fix a number $r$ and ask which equations have $r$ as a root. The condition is $r^2 + pr + q = 0$, which rearranges to

$$q = -r\,p - r^2.$$

That is a straight line in the $(p, q)$ plane. So “the equations that $r$ solves” is not a scattered set but a line, one line for each candidate root, and the plane is crossed by a whole family of them.

Every one of those lines touches the parabola, once, and stays on the lower side of it. The proof is a completed square. The line for $r$ is below the curve at $p$ exactly when $-rp - r^2 \le p^2/4$, and moving everything to one side gives

$$\frac{p^2}{4} + rp + r^2 = \left(\frac{p}{2} + r\right)^2 \ge 0,$$

with equality only at $p = -2r$. So the line meets the curve at the single point $(-2r, r^2)$ — which is the perfect square $(x - r)^2$, the one equation that has $r$ as a double root — and everywhere else it passes underneath.

Read the figure with that in hand and the discriminant explains itself. A point below the parabola has two of these tangent lines passing through it, because from any point outside a convex curve there are exactly two tangents. Those two lines are its two roots. A point on the parabola has one tangent line through it, the one that touches there: one repeated root. A point above the parabola is inside the bowl, and no tangent line reaches it: no real root. The count of real roots is the count of tangents that can be drawn to one fixed curve from the point, and the discriminant is the condition for being outside that curve.

This is a genuinely different way of saying what a root is. In the usual picture a root is a crossing point on the $x$-axis of one particular graph. Here a root is a line, the equation is a point, and solving means finding the lines through the point. The parabola is what geometers call the envelope of the family: the curve that every member touches and that the family, drawn densely, outlines without anybody drawing it. The dashed curve in the figure was drawn deliberately, but thirteen lines are already enough to make it visible on its own.

The plane of roots, folded in half #

The tangent-line picture has a second reading, which explains why the boundary has to be a curve along which something is creased.

Start from the roots instead of the equation. A pair of real roots $(r, s)$ is a point of another plane, and the equation it produces is $(x - r)(x - s) = x^2 - (r + s)x + rs$, which is the point $(p, q) = (-(r + s),\ rs)$. So there is a map from the plane of roots to the plane of equations, and the whole of this essay’s content is what that map does.

Two features of the map are visible at once. First, it is two-to-one: the pair $(r, s)$ and the swapped pair $(s, r)$ produce the same equation, because an equation does not know which of its roots was listed first. The shaded half of the left panel, where $r < s$, already reaches every equation with two distinct real roots, and the unshaded half covers the same region a second time.

Second, the diagonal $r = s$ is where the two copies meet, and it maps to the parabola. So the region under the parabola is the plane of roots folded along its diagonal: the two halves laid on top of each other, with the fold line becoming the boundary. The parabola is not merely where the discriminant happens to vanish; it is the crease of a folded sheet, and a crease is the edge of the image of a fold for the same reason the edge of a folded tablecloth is where it doubles back.

That also explains what the discriminant is doing algebraically. The sum $r + s$ and the product $rs$ are symmetric — unchanged by the swap — and so is $(r - s)^2$. It is the simplest symmetric quantity that vanishes on the fold, which is why it is the one that decides which side of the crease a point is on. The formula $p^2 - 4q$ is $(r - s)^2$ written in the only coordinates the equation has.

How often a quadratic has real roots #

Once the answer is a region, a question that sounds like probability becomes a question about area.

Choose $p$ and $q$ at random, each spread evenly between $-1$ and $1$. The equation $x^2 + px + q$ is then a random point of a two-by-two square, and it has real roots exactly when the point lies under the parabola. Inside that square the parabola barely rises — it reaches only $1/4$ at the corners — so the region below it is the whole lower half of the square plus a thin lens above the middle line. Its area is

$$\int_{-1}^{1} \left(\frac{p^2}{4} + 1\right) dp = \frac{1}{6} + 2 = \frac{13}{6},$$

out of a square of area $4$, so the chance is $13/24$, a little over a half.

The general equation $ax^2 + bx + c$, with all three coefficients random in the same way, does better, and the reason is a small piece of geometry hiding inside the algebra. Real roots need $b^2 \ge 4ac$. Half the time $a$ and $c$ have opposite signs, the product $ac$ is negative, and the inequality holds whatever $b$ is. The other half of the time it holds only when $|ac|$ is small compared with $b^2/4$, which happens with a probability that has a logarithm in it — the chance that a product of two evenly spread numbers falls below $t$ is $t - t \ln t$. Averaging over $b$ gives $(41 + 6\ln 2)/72 \approx 0.627$.

The two numbers answer genuinely different questions, and the difference is instructive. Fixing the leading coefficient at $1$ is not a harmless normalisation for a probability: dividing a random $ax^2 + bx + c$ through by $a$ produces a monic equation whose coefficients are no longer evenly spread, because dividing by a small $a$ throws them far out into the plane — and far out, the plane is mostly under the parabola. The shape of the region does not change; the way points are scattered over it does, and a probability depends on both — which is the same lesson dropping needles to estimate π teaches from the other side, where the answer is only as good as the stated way of throwing.

The cubic’s curve has a point on it #

The same construction works one degree up and produces a picture with a feature the quadratic does not have.

A cubic can always be shifted so that its $x^2$ term vanishes, which leaves $x^3 + px + q$ — two coefficients again, so again a plane. A number $r$ is a root when $r^3 + pr + q = 0$, which is once more a straight line, $q = -r^3 - rp$. The family of lines, one per candidate root, again outlines a curve, and this time the curve is

$$4p^3 + 27q^2 = 0.$$

That curve is not smooth. Its two branches come in from the left, $q = \pm\sqrt{-4p^3/27}$, and meet at the origin in a cusp — a sharp point where the curve’s direction reverses. Inside the pointed region, where $p$ is negative and $q$ small, three tangent lines reach each point and the cubic has three real roots; outside, only one line reaches and the cubic has one real root and a pair of complex ones.

The roots in that figure were found without using the curve at all — by locating sign changes of the cubic along the line and halving the interval until the root was pinned — and the counts came out as the curve predicts. That separation matters, since otherwise the picture would be checking a formula against itself.

The cusp is where all three roots coincide. The equation at the origin is $x^3$, whose only root is $0$, counted three times. Moving away from the origin along either branch separates one root from the other two, which stay together; moving into the pointed region separates all three. So the curve has more structure than the quadratic’s parabola: its smooth parts are where exactly two roots meet, and the one sharp point is where three do. The quadratic’s parabola has no such point because a quadratic has only two roots to lose.

What the tangent lines are, seen from further away #

The construction in this essay is older than it looks and has a name. Replacing a curve by the family of its tangent lines, and a point by the family of lines through it, is projective duality, and it is the move the function seen from its tangents makes for convex functions: describe the object by its tangents rather than its points, and a second description appears with its own geometry.

Here the duality runs in a specific direction. The curve of repeated-root equations is traced by points; its tangent lines are the equations sharing one root. So a question about roots — how many, and when do they merge — has become a question about a curve and its tangents, and the answers are the ones school geometry gives about tangents from an external point.

The pattern continues upward and becomes one of the organising ideas of modern geometry. For polynomials of degree $n$ the space of equations has $n$ dimensions, the equations with a repeated root form a hypersurface in it, and that hypersurface has a hierarchy of worse and worse singularities where more and more roots coincide. For the quartic the surface has creases and sharp points, and its shape — called the swallowtail — is one of the standard figures of singularity theory. The parabola here and the cusp above are the first two members of the same list. None of it needs the roots to have a formula: past degree four they generally do not, because a group of sixty symmetries will not come apart, and the surface of colliding roots is exactly as well defined as it was for the quadratic.

There is also a sharper consequence closer to home. The cubic’s cusped curve is exactly where the formula for solving a cubic runs into trouble. Inside the pointed region the three roots are real, and yet Cardano’s formula reaches them only by passing through square roots of negative numbers; the cusp region is the casus irreducibilis drawn as a place. It is also the region where a marked ruler trisects an angle: the cubic that trisection solves has three real roots, so it lives inside the cusp. That the difficulty should occupy a region with a sharp point on its boundary is not a coincidence: the square root inside the formula is the square root of a positive multiple of $4p^3 + 27q^2$, so it is the square root of a negative number exactly inside the pointed region, and the curve is where that quantity changes sign.

Where the plane stops being honest #

Three limitations, each specific.

The plane holds only the real equations. Every point drawn has real coefficients, and the whole region above the parabola is labelled “no real root” as though nothing were there. There is something there — each such equation has two complex roots — but they cannot be drawn as lines in this plane, because a line $q = -rp - r^2$ with $r$ complex is not a line of real points. The region above the curve is where the picture runs out of room rather than where the roots run out of existence, and the plane where multiplying is turning is where they live instead.

The drawn lines are a sample. Thirteen lines stand in for a continuum, one for every real number. The envelope appears because the lines are dense enough for the eye to fill in, but a point that happens to fall between two drawn lines is not thereby on no line: through every point below the curve pass exactly two tangent lines, almost always neither of them drawn. The figure shows the shape of the family; the claim that there are exactly two lines through each point comes from the algebra, not from counting lines in the picture.

The probabilities depend on a choice nobody forced. The share $13/24$ belongs to coefficients spread evenly over $[-1, 1]$. Spread them evenly over $[-10, 10]$ instead and the bowl above the origin is a much smaller part of the square, so the share rises towards one; spread them by a bell curve and it becomes something else again. There is no uniform distribution over the whole plane, and so no answer to “how often does a quadratic have real roots” without first saying how the quadratic was chosen.

Still open: the counting at higher degree #

For quadratics and cubics everything here is settled, and the plane of equations is the whole story. At higher degree the easy statements survive and the counting questions become research.

The expected number of real roots of a polynomial of degree $n$ whose coefficients are independent standard bell-curve variables is known precisely in the limit: it grows like $\frac{2}{\pi}\ln n$, a result of Kac from 1943, so a random polynomial of degree a million has only about nine real roots on average. What is much less settled is the distribution around that average — how likely a random polynomial is to have no real roots at all, or all of them real. Dembo, Poonen, Shao and Zeitouni showed in 2002 that the chance of no real roots at large even degree falls off like $n$ to a fixed negative power, and could only bracket the power; simulations suggested $3/4$, and an exact value of $3/4$ was obtained only in 2018, by Poplavskyi and Schehr, through a calculation from statistical physics about how long a diffusing surface stays positive.

The geometry has its own open corners. The hypersurfaces of equations with repeated roots — the discriminant hypersurfaces — are completely understood in their local shapes, but the global counting of how many distinct types of real root configuration occur, and how the regions separating them fit together, becomes combinatorially heavy quickly, and for systems of several polynomials in several unknowns it is an active subject. The next step up this path is the square completed in an exponent, where the same algebra that produced the parabola here computes the area under a bell curve — and every other bell curve besides.

A root is a line, a repeated root is a tangency #

The discriminant is usually met as a formula to memorise and a sign to inspect. Laid out as a plane, it is a curve, and the curve is not arbitrary: it is the set of perfect squares, the crease where the plane of root pairs folds onto itself, and the envelope of the lines that represent individual roots. Each of those three descriptions accounts for the other two.

What makes the picture worth having is that it converts a statement about numbers into one about position. “Two real roots” becomes “outside the curve, where two tangents reach”. “The roots collide” becomes “the point crosses the crease”. And a question that has no obvious geometric content — how often is a quadratic solvable over the reals — becomes the area of a region under a parabola, answered in two lines of calculus, with $13/24$ at the end of them.