ch1.4.13-quartic-equations

Ch 1.4.13 — Of the Resolution of Equations of the Fourth Degree

Summary: Euler opens the treatment of quartics with pure and incomplete cases, extends Vieta’s formulas and the rational root theorem to degree 4, introduces Descartes’ sign rule, and handles two special families of symmetric-coefficient equations.

Sources: chapter-1.4.13

Last updated: 2026-05-03

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Pure quartics (§750–752)

A pure (incomplete) quartic has the form $x^4=f$. Euler extracts the biquadrate root in two steps:

$x^2=\pm \sqrt{f},x=\pm \sqrt{\pm \sqrt{f}}$

There are always four roots. For example, $x^4=2401$:

• $x^2=49\Rightarrow x=\pm 7$
• $x^2=-49\Rightarrow x=\pm 7\sqrt{-1}$

Two roots are real and two are imaginary, unless $f<0$, in which case all four are imaginary.

Incomplete quartics (§753)

When the second and fourth terms vanish, the form is $x^4+f{x}^2+g=0$. Setting $y=x^2$ reduces it to a quadratic in $y$:

$y=\frac{-f\pm \sqrt{f^2-4g}}{2}$

and then $x=\pm \sqrt{y}$, giving all four roots from the two values of $y$.

Vieta’s formulas for quartics (§754–755)

The general quartic factors as $(x-p)(x-q)(x-r)(x-s)=0$, expanding to:

$x^4-(p+q+r+s)x^3+(pq+pr+ps+qr+qs+rs)x^2-(pqr+pqs+prs+qrs)x+pqrs=0$

Reading off the coefficients of $x^4+a{x}^3+b{x}^2+cx+d=0$:

Coefficient	Symmetric function of roots
$-a$	$p+q+r+s$ (sum)
$b$	$\sum pq$ (sum of products two at a time)
$-c$	$\sum pqr$ (sum of products three at a time)
$d$	$pqrs$ (product of all four)

This extends vieta-formulas from degree 2 and 3 to degree 4.

Rational root theorem for quartics (§756–757)

Since $d=pqrs$, any rational root must divide the constant term $d$ (source: §756). This mirrors the rational-root-theorem for cubics.

To apply it: if the equation has fractional coefficients, first clear denominators by substituting $x=y/n$ for a suitable integer $n$ that makes all coefficients integers with leading coefficient 1.

Once a rational root $r$ is found, divide the quartic by $(x-r)$ to obtain a cubic solvable by cardanos-rule.

Descartes’ sign rule (§758)

Given an integer-coefficient quartic with the second term arranged left to right:

• The number of positive roots equals the number of sign changes (+ to − or − to +) between consecutive nonzero terms.
• The number of negative roots equals the number of consecutive same signs (successions).

For example, $x^4-3x^3+12x^2-162x+72=0$ has four sign changes and zero successions, so all four roots are positive — no need to test negative divisors (§758). See descartes-sign-rule.

Worked example (§759)

$x^4+2x^3-7x^2-8x+12=0$

Two sign changes, two successions → two positive and two negative roots. Divisors of 12: 1, 2, 3, 4, 6, 12.

Testing positive: $x=1$ works; $x=2$ works. Testing negative: $x=-3$ works; $x=-4$ works.

All four roots: $x=1,2,-3,-4$.

Special case I: palindromic equations (§761–762)

When the coefficients read the same forwards and backwards — $x^4+ma{x}^3+n{a}^2{x}^2+m{a}^{3}x+a^4=0$ — the quartic factors into two quadratics:

$(x^2+pax+a^2)(x^2+qax+a^2)=0$

where $p+q=m$ and $pq=n-2$. This gives $p-q=\sqrt{m^2-4n+8}$, so:

$p=\frac{m+\sqrt{m^2-4n+8}}{2},q=\frac{m-\sqrt{m^2-4n+8}}{2}$

Each quadratic then yields two roots $x=-\frac{pa}{2}\pm \frac{a}{2}\sqrt{p^2-4}$.

Example (§762): $x^4-4x^3-3x^2-4x+1=0$ (here $a=1$, $m=-4$, $n=-3$). Then $\sqrt{m^2-4n+8}=6$, so $p=1$, $q=-5$. The four roots are $\frac{-1\pm \sqrt{-3}}{2}$ (imaginary) and $\frac{5\pm \sqrt{21}}{2}$ (real).

Special case II: anti-palindromic equations (§763–764)

When the second and fourth terms have opposite sign from the palindromic case: $x^4+ma{x}^3+n{a}^2{x}^2-m{a}^{3}x+a^4=0$. This factors as $(x^2+pax-a^2)(x^2+qax-a^2)=0$, leading to $p-q=\sqrt{m^2-4n-8}$ and four roots $x=-\frac{pa}{2}\pm \frac{a}{2}\sqrt{p^2+4}$ (or similarly for $q$).

Example (§764): $x^4-3\cdot 2x^3+3\cdot 8x+16=0$ has roots $1\pm \sqrt{5}$ and $2\pm \sqrt{8}$.

Related pages

• quartic-equations
• vieta-formulas
• rational-root-theorem
• descartes-sign-rule
• imaginary-numbers
• ch1.4.14-bombelli-rule
• ch1.4.15-new-method-quartic
• cardanos-rule
• equations