Fourth degree equation of general type

ax ⁴ + bx ³ + cx ² + dx + e = 0. - Equation of fourth degree of general form.

1. Reduction to an incomplete equation

The fourth degree equation is generally reduced to an incomplete equation.

y ⁴ + py ² + qy + r = 0

by changing variables

x = y - b / 4a

2. Decarte-Euler Decision

The roots of an incomplete fourth degree equation are determined by the formulas:



where z ₁ , z ₂ , z ₃ are the roots of the cubic equation

z ³ + 2pz ² + (p ² - 4r) - q ² = 0,

which is called the resolvent of the original equation. The signs of the roots in it are chosen in such a way that equality holds:



The roots of the incomplete initial equation are determined by the roots of the cubic resolvent according to the table:

The connection between the roots of an incomplete fourth-degree equation and the roots of the resolvent

Cubic resolvent	Fourth degree equation
All roots are valid and positive (*)	Four valid roots
All roots are valid, they are positive and two are negative (*)	Two pairs of complex conjugate roots
One root is positive, two complex conjugates	Two real and two complex conjugate roots

(*) - by the Viet theorem, the product of the roots z ₁ , z ₂ , z ₃ = q ²

3. Ferrari Solution

Suppose z ₀ is one of the roots of an auxiliary cubic equation.

z ³ + 2pz ² + (p ² - 4r) - q ² = 0,

Then the fourth roots of an incomplete fourth degree equation are found by solving two quadratic equations