Numbers & Functions · 4 Polynomials, Factoring, and the Quadratic Formula

4.8 The discriminant: a fork in the road

Look hard at the one piece sitting under the square root: b2 −4ac. This little expression has its own name — the discriminant — because it discriminates, in the old sense of “tells things apart.” Its sign alone tells you what kind of roots you’ll get: • b2 −4ac > 0: two different real roots. The graph crosses the axis twice. • b2 −4ac = 0: one repeated root. The graph just kisses the axis and turns back.

• b2 −4ac < 0: you’re asked for the square root of a negative number. Stop right there. We are at the exact spot I promised back in Chapter 1. The square-root machine jams on negative inputs. So when the discriminant comes out negative, the formula is demanding a number that, among the reals, simply does not exist. For two and a half thousand years, this is where mathematicians stopped. But you know this story by now. Every time an operation has jammed, we invented the numbers that made it work.

x b 2 −4ac > 0 two real roots x b 2 −4ac = 0 one repeated root x b 2 −4ac < 0 no real roots yet The discriminant decides how many times the curve meets the axis Figure 4.4. The discriminant fork. The sign of b2 −4ac decides whether the parabola crosses the axis twice, kisses it once, or floats clear of it (no real roots — yet).

The negative discriminant is the door, and in Chapter 8 we walk through it. When we do, every quadratic — every single one — will turn out to have exactly two roots.

4.9 A glimpse of deep water

If you’ve been paying attention, a question should be itching at you. We have a formula for the quadratic. So what about degree 3? Is there a cubic formula? And a quartic, and a quintic, forever?

Degrees 3 and 4: yes, but at a terrible price. There is a cubic formula, found in Renaissance Italy in the 1500s amid a soap opera of broken oaths and public duels. Tartaglia worked out how to solve cubics and told Cardano under a sworn vow of secrecy — which Cardano broke, publishing it in 1545. His student Ferrari cracked the quartic too. The cubic and quartic formulas are real — but they are monsters, pages of nested roots, almost useless by hand. Degree 5: no. And not “we couldn’t find it” — “it is impossible.” In the early 1800s a young Norwegian, Niels Henrik Abel, proved there is no formula for the general quintic. Not a complicated one — none can exist, ever, built from +, −, ×, ÷ and roots. The pattern that held for degrees 2, 3, 4 simply shatters at 5.

Why does it break? Answering that required a French teenager, Évariste Polynomials, Factoring, and the Quadratic Formula Galois, to invent an entirely new kind of mathematics — not about numbers at all, but about symmetry: the ways the roots can be shuffled without anything noticing. His tool is now called group theory, and groups turned out to describe symmetry everywhere — in crystals, in physics, in the codes that keep your messages private. Galois wrote down the heart of these ideas the night before he was killed in a duel, at the age of twenty.

The point of telling you now: mathematics is far bigger than the formulas in this book. Underneath the everyday machinery there are vast, strange, beautiful structures that often look “useless” when discovered and turn out to hold up the modern world.

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