Cubic Equations

Summary: Euler’s treatment of equations of the third degree — pure, complete, and general forms — including the three-root theorem, Vieta’s relations for cubics, the rational root theorem, the sign rule, and Cardan’s formula.

Sources: chapter-1.4.10, chapter-1.4.11, chapter-1.4.12

Last updated: 2026-05-03

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Classification

Type	Form	Method
Pure (incomplete)	$x^3=a$	Direct cube-root extraction
Complete	$a{x}^3\pm b{x}^2\pm cx\pm d=0$	Rational root trial; then factor
Depressed (no $x^2$)	$x^3=fx+g$	Cardan’s formula directly
General complete	$x^3+a{x}^2+bx+c=0$	Substitute $x=y-a/3$, then Cardan

Three Roots

Every cubic equation has exactly three roots (source: chapter-1.4.10, §712–713). For the pure cubic $x^3=a$ with real cube root $c=\sqrt[3]{a}$:

$x_1=c,x_{2,3}=\frac{-1\pm \sqrt{-3}}{2}\cdot c$

Only $x_1$ is real; $x_2$ and $x_3$ are imaginary-numbers. This is the first case where Euler exhibits three roots explicitly.

More generally, for any cubic with roots $p,q,r$:

$(x-p)(x-q)(x-r)=x^3-(p+q+r)x^2+(pq+pr+qr)x-pqr$

Vieta’s Relations for Cubics

Comparing $x^3-a{x}^2+bx-c=0$ with the factored form gives (source: chapter-1.4.11, §722):

$p+q+r=a,pq+pr+qr=b,pqr=c$

These extend vieta-formulas from degree 2 to degree 3.

Rational Root Theorem

Any rational root of an integer-coefficient monic cubic $x^3-a{x}^2+bx-c=0$ must be an integer divisor of the constant term $c$ (source: chapter-1.4.11, §722). This follows from $c=pqr$.

After testing divisors, if a root $r$ is found, divide by $(x-r)$ to reduce to a quadratic-equations problem.

Sign Rule (Descartes)

In a cubic (source: chapter-1.4.11, §725):

• The number of positive roots equals the number of sign changes between consecutive terms.
• The number of negative roots equals the number of times the same sign repeats consecutively.

This guides which divisors (positive or negative) to try first.

Cardan’s Formula

For the depressed cubic $x^3=fx+g$, one root is (source: chapter-1.4.12, §741):

$x=\sqrt[3]{\frac{g+\Delta }{2}}+\sqrt[3]{\frac{g-\Delta }{2}},\Delta =\sqrt{g^2-\frac{4}{27}{f}^3}$

See cardanos-rule for derivation and examples. For the general cubic, first apply $x=y-a/3$ to eliminate the $x^2$ term, then use this formula.

When the cubic has no rational root, the nested radicals in Cardan’s formula cannot be simplified; this is the casus irreducibilis (source: chapter-1.4.12, §749).

Related pages

• ch1.4.10-pure-cubic-equations
• ch1.4.11-complete-cubic-equations
• ch1.4.12-cardanos-rule
• ch2.0.8-cubic-surd-rationalization
• ch2.0.10-cubic-formula-as-cube
• cardanos-rule
• rational-root-theorem
• vieta-formulas
• quadratic-equations
• imaginary-numbers
• completing-the-square
• sum-of-two-cubes