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When you write ax² + bx + c = 0 and solve it once for every possible a, b and c, you are using an idea that is barely 435 years old. Before François Viète, algebra books mostly worked with particular numbers and described methods in words. This post covers who Viète was, what his letters changed, and how his link between roots and coefficients still shortens your working.
Who was François Viète?
Viète was born in 1540 in Fontenay-le-Comte in western France and died in Paris on 13 December 1603, according to his MacTutor biography. He took a law degree at Poitiers in 1560 and spent most of his life as a lawyer and royal counsellor, never as a paid mathematician.
His most productive mathematical years came from bad luck. In 1584, as a known Huguenot, he was banished from the royal court by political enemies and spent about five years at Beauvoir-sur-Mer, where, MacTutor says, his most important mathematics was done. Back in royal service, he broke a Spanish cipher for Henry IV and sent him the fully decoded letter on 15 March 1590. In 1593 the mathematician Roomen posed an equation of degree 45, and a visiting ambassador claimed no Frenchman could solve it; Viète solved it by spotting a hidden trigonometric relation. In the preface to his reply he called himself someone "who do not profess to be a mathematician, but who, whenever there is leisure, delight in mathematical studies" (as translated by MacTutor).
What changed in 1591?
In *In artem analyticem isagoge* ("Introduction to the analytic art"), printed at Tours in 1591, Viète used letters for both unknown and known quantities. A first edition is scanned on the Internet Archive. The MacTutor page on the earliest symbols for variables gives his rule in translation: unknowns get A or another vowel (E, I, O, U, Y), and given quantities get B, G, D or other consonants. Descartes later switched to x, y, z for unknowns and a, b, c for knowns, the convention you use today.
Letters for unknowns were not new. Letters for the given numbers were. Once the coefficients are letters, a single piece of working proves a result for every equation of that type. MacTutor also notes that the word "coefficient" is due to Viète.
His algebra still had limits. He insisted on a "law of homogeneous quantities": every term had to have the same dimension, so he wrote equations like A³ + B²A = B²Z rather than x³ + x = 1, because adding a cube to a line made no geometric sense to him.
How do roots and coefficients connect?
MacTutor says Viète knew the connection between the positive roots of an equation and its coefficients. In modern form, your NCERT Class 10 chapter on polynomials (section 2.3) states it for ax² + bx + c: the sum of the zeroes is −b/a and the product is c/a. The JEE (Main) 2026 syllabus lists "relations between roots and coefficients" in Unit 2. Check the syllabus for your exam year.
Worked example: using sum and product
Find the roots of x² − 7x + 10 = 0 without the formula. Here a = 1, b = −7, c = 10. Sum = 7, product = 10. Two numbers with sum 7 and product 10 are 2 and 5. Check: 2² − 14 + 10 = 0 and 5² − 35 + 10 = 0.
Build an equation from its roots. Roots 3 and −4 have sum −1 and product −12, so the equation is x² − (−1)x + (−12) = 0, that is x² + x − 12 = 0.
A cubic. For ax³ + bx² + cx + d with roots α, β, γ: α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a. For x³ − 6x² + 11x − 6, the roots 1, 2, 3 give sum 6, pairwise sum 2 + 3 + 6 = 11 and product 6, matching −(−6)/1, 11/1 and −(−6)/1.
Common mistakes with these relations
Dropping the minus sign. The sum is −b/a, not b/a. In x² − 7x + 10 the sum is +7.
Forgetting to divide by a. For 2x² − 8x + 6 the sum is 8/2 = 4, not 8.
Wrong sign on a cubic's product. For a cubic the product is −d/a; for a quadratic it is +c/a.
Stopping at real roots. The relations hold for complex roots too, which is why JEE asks about them even when the roots are not real.
Our guides to polynomials in Class 10 and quadratic equations and their roots have more practice.
Did Viète also work on π?
Yes. MacTutor reports that he calculated π to 10 decimal places using a polygon with 6 × 2¹⁶ = 393,216 sides, and wrote π as an infinite product, the earliest known infinite expression for π. Sixty years later John Wallis found a different product, covered in our post on John Wallis and his infinite product for π.
What can a student take from Viète?
Viète did his finest work in spare time, and his main contribution was a way of writing, not a single theorem. Good notation lets you see a whole family of problems at once. Each time you replace a number with a letter and the working still goes through, you have proved something general.
If you can apply sum and product but cannot see why they hold, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer.
Practice routine for this week:
Write five quadratics with integer roots and recover the roots from sum and product alone.
Form the equation whose roots are 2α and 2β, given x² − 5x + 6 = 0, and check by solving.
Prove sum = −b/a by expanding a(x − α)(x − β) yourself.
Tonight, take one formula you use often and write it with letters for every number, then check it still works on two examples.
