Quartic Equations

Summary: Euler’s treatment of equations of the fourth degree — four roots, Vieta’s relations, the rational root theorem, Descartes’ sign rule, and two general solution methods (Bombelli and the radical ansatz).

Sources: chapter-1.4.13, chapter-1.4.14, chapter-1.4.15

Last updated: 2026-05-03

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Classification

Type	Form	Method
Pure	$x^4=f$	Two successive square roots; four roots $\pm \sqrt{\pm \sqrt{f}}$
Incomplete	$x^4+f{x}^2+g=0$	Substitute $y=x^2$; reduce to quadratic
Palindromic	Coefficients symmetric	Factor into two quadratics
General	$x^4+a{x}^3+b{x}^2+cx+d=0$	Bombelli or radical-ansatz method

Four Roots

Every quartic has exactly four roots (source: chapter-1.4.13, §752). They arise from the factored form:

$(x-p)(x-q)(x-r)(x-s)=0$

Some roots may be imaginary; Euler shows that the pure quartic $x^4=2401$ has $x=\pm 7$ (real) and $x=\pm 7\sqrt{-1}$ (imaginary). See ch1.4.13-quartic-equations.

Vieta’s Relations

For $x^4+a{x}^3+b{x}^2+cx+d=0$ with roots $p,q,r,s$ (source: chapter-1.4.13, §754–755):

$p+q+r+s=-a,\sum pq=b,\sum pqr=-c,pqrs=d$

See vieta-formulas for the general framework.

Rational Root Theorem

Any rational root must divide the constant term $d$ (source: §756), since $d=pqrs$. This extends rational-root-theorem from cubics to quartics. Fractional coefficients must be cleared first (substitute $x=y/n$).

Descartes’ Sign Rule

The number of positive roots equals the number of sign changes between consecutive terms; the number of negative roots equals the number of same-sign successions (source: §758). See descartes-sign-rule.

Solution Methods

Bombelli’s Method (§766–768, Chapter XIV)

Write $x^4+a{x}^3+b{x}^2+cx+d=(x^2+\frac{a}{2}x+p{)}^2-(qx+r{)}^2$. Matching coefficients produces the Bombelli cubic in $p$:

$8p^3-4b{p}^2+(2ac-8d)p-a^{2}d+4bd-c^2=0$

Solve this cubic for $p$; recover $q$ and $r$; then split the quartic into two quadratics. See bombelli-rule and ch1.4.14-bombelli-rule.

Euler’s Radical Ansatz (§774–778, Chapter XV)

Assume $x=\sqrt{p}+\sqrt{q}+\sqrt{r}$ where $p,q,r$ are roots of an auxiliary cubic $z^3-f{z}^2+gz-h=0$. For the depressed quartic $x^4-a{x}^2-bx-c=0$:

$f=\frac{a}{2},g=\frac{a^2}{16}+\frac{c}{4},h=\frac{b^2}{64}$

The four roots are all sign combinations of $\pm \sqrt{p}\pm \sqrt{q}\pm \sqrt{r}$ consistent with $\sqrt{pqr}=b/8$. See ch1.4.15-new-method-quartic.

Removing the cubic term

Substitute $y=x-a/4$ in $y^4+a{y}^3+\cdots =0$ to eliminate the $x^3$ term before applying either method (source: §779).

Limits of Algebraic Methods

Euler notes (§780) that no general formula is known for equations of degree 5 or higher. Approximation methods must be used instead; see approximation-methods and ch1.4.16-approximation.

Related pages

• ch1.4.13-quartic-equations
• ch1.4.14-bombelli-rule
• ch1.4.15-new-method-quartic
• ch2.0.9-quartic-surd-rationalization
• bombelli-rule
• descartes-sign-rule
• vieta-formulas
• rational-root-theorem
• cubic-equations
• quadratic-equations
• imaginary-numbers
• equations