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In 1540, the Italian mathematician Lodovico Ferrari discovered how to solve the quartic equation by showing that it can be reduced to solving a cubic equation. His method, which relies on finding a "resolvent cubic" to complete the square twice, was eventually published in Gerolamo Cardano's *Ars Magna* in 1545.
Why was Ferrari's quartic solution delayed?
Lodovico Ferrari's path to mathematical history was unusual. He arrived at Gerolamo Cardano's house in Milan in 1536 to work as a servant. Cardano soon recognised the boy's mathematical ability and promoted him to secretary, actively mentoring him in mathematics.
By 1540, at the age of 18, Ferrari had solved the general quartic (fourth-degree) equation. However, his method relied entirely on being able to solve cubic (third-degree) equations as an intermediate step. At the time, Cardano had promised Niccolò Tartaglia that he would keep Tartaglia's newly discovered cubic solution a secret. Because Ferrari's solution required Tartaglia's algebra to work, the quartic solution could not be published immediately.
It was finally shared with the world five years later in Cardano's book *Ars Magna* in 1545, once Cardano found an independent source for the cubic solution.
How did Lodovico Ferrari solve the quartic equation?
Ferrari's approach was not in finding a direct arithmetic formula, but in transforming the structure of the problem. He realised that a quartic equation could be rewritten as an equality between two perfect squares, which could then simply be square-rooted and solved as ordinary quadratic equations.
Here is the framework of Ferrari's method for solving a general quartic equation, Ax⁴ + Bx³ + Cx² + Dx + E = 0:
Depress the quartic: Substitute x = u − B/(4A) to eliminate the x³ term entirely. This leaves a "depressed" quartic of the form u⁴ + au² + bu + c = 0.
Isolate the fourth power: Rearrange the terms so the highest power is alone on the left side:
u⁴ = −au² − bu − c
Complete the square on the left: Add 2yu² + y² to both sides, introducing a new, carefully chosen unknown variable y. The left side now factors perfectly into a square:
(u² + y)² = (2y − a)u² − bu + (y² − c)
Force the right side to be a perfect square: The right side is now a quadratic expression in u. For it to be a perfect square, its discriminant (the b² − 4ac part of the standard quadratic formula) must equal zero.
Solve the quadratics: Once you find the correct value for y, both sides of the equation are perfect squares. Taking the square root of both sides leaves two standard quadratic equations. Solving those yields the four roots of the original quartic.
(If you want to read about the cubic solution he relied on, see Niccolò Tartaglia and the cubic rule he hid in a poem, or review the algebraic rules for When two quadratics share a root: the condition and its use).
What is a resolvent cubic?
In step 4 of his framework, Ferrari required the discriminant of the right side to be exactly zero: (−b)² − 4(2y − a)(y² − c) = 0
When you expand this requirement, it creates a new polynomial equation entirely in terms of y: 8y³ − 4ay² − 8cy + (4ac − b²) = 0
This third-degree polynomial is known as Ferrari's "resolvent cubic". By setting the discriminant to zero, Ferrari proved that finding the parameters to solve a quartic equation inevitably reduces down to finding the root of a cubic equation.
How did Ferrari's career end?
After the publication of *Ars Magna*, Ferrari's reputation grew rapidly. His academic career flourished; having started teaching mathematics at just 18, he succeeded his mentor Cardano at the Piatti Foundation in 1541.
His fame peaked in 1548 when he defeated Tartaglia in a high-profile, public mathematical debate in Milan. The victory established his status as a leading mathematician in Italy. He was eventually appointed as a professor of mathematics at the University of Bologna in 1565, the same year he died.
Breaking down complex mathematical problems
Ferrari's strategy—reducing a completely new and intractable problem into an older, solvable one—is one of the most powerful active learning habits a student can build. Often, the hardest part of advanced mathematics or physics is simply not knowing where to start.
If you are stuck on a multi-step numerical problem, Fermi's AI tutor can work out what you already know and build a plan for the session.
When you face a daunting question in your exam prep, try to strip away the highest-level complication first, just as Ferrari depressed his equation to remove the cubic term. If you are unsure how the logical steps connect, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer. Just as Lodovico Ferrari broke the fourth-degree polynomial into a cubic and then into quadratics, you can learn to turn a complex 12-mark exam question into a series of familiar, manageable steps.
