Ferrari's Solution to the quartic equation

In today's blog, I go through Lodovico Ferrari's solution of the general quartic equation. In my next blog, I will present a proof of Ferrari's method.

The content in today's blog is taken from Jean-Pierre Tignol's Galois' Theory of Algebraic Equations.

(1) Ferrari started out with a quadratic equation of the following form:

x⁴ + bx³ + cx² + dx + e = 0

(2) Then, he borrowed a method from Girolamo Cardano (see here) and set y = x + (b/4) which means x = y - (b/4).

(3) Substituting x = y - (b/4) gives us:

(y - [b/4])⁴ + b(y - [b/4])³ + c(y - [b/4])² + d(y - [b/4]]) + e = 0

Working this through (see details here), gives us:

y⁴ + py² + qy + r = 0

where:

p = c - 6(b/4)²

q = d - (b/2)c + (b/2)³

r = e - (b/4)c + (b/4)²c - 3(b/4)⁴

(4) Now if we move the q and r term over we get:

y⁴ + py² = -qy -r

This helps us because if we add (p/2)² to both sides, we can complete the square to get:

(y² + p/2)² = -qy -r + (p/2)²

(5) Now, if we add a value u to our square, we get the following:

(y² + p/2 + u)² = -qy -r + (p/2)² + 2uy² + pu + u²

This may not appear to help us but it gives us a possibility for completing the square on the other side.

(6) For example, if we could find a value u such that:

-qy -r + (p/2)² + 2uy² + pu + u² = ([√2uy - q/(2√2u)]²

Then we have found a solution (see step #11 if this point is not clear)

(7) Now:

([√2uy - q/(2√2u)]² = 2u²y² - 2*(√2uy)(q/(2√2u)) + q²/(8u) =

= 2u²y² - q/u + q²/(8u)

(8) So this equation works only if:

-qy -r + (p/2)² + 2uy² + pu + u² = 2uy² - qy + q²/(8u)

That is if:

-r + (p/2)² + pu + u² = q²/(8u)

(9) But if we multiply 8u to all sides and then subtract by q², we get:

8u³ + 8pu² + (2p² - 8r)u - q² = 0

(10) This is the general cubic equation which we can solve using Cardano's formula so we know that u can take on this value.

(11) This then gives us:

(y² + p/2 + u)² = ([√2uy - q/(2√2u)]².

(12) This can we solve using the quadratic equation. The four solutions can be found from:

(y² + p/2 + u)² = ± [√2uy - q/(2√2u)]

I should probably note that we assumed that u ≠ 0. I will talk about the situation were u = 0 in my next blog where I provide a proof for Ferrari's solution.

References

• Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, 2002