ch1.4.11-complete-cubic-equations

Ch 1.4.11 — Of the Resolution of Complete Equations of the Third Degree

Summary: Resolves complete cubics $a{x}^3\pm b{x}^2\pm cx\pm d=0$ using the factored form, Vieta’s relations for three roots, the rational root theorem, a sign rule for positive/negative roots, and six worked word problems.

Sources: chapter-1.4.11

Last updated: 2026-05-03

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Definition (§719)

A cubic is complete when all four terms are present:

$a{x}^3\pm b{x}^2\pm cx\pm d=0$

Every such equation has exactly three roots (§719, by analogy with the pure cubic result of the preceding chapter).

Factored Form and Vieta’s Relations for Cubics (§720–722)

A complete cubic can be written as the product of three linear factors:

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

Expanding gives $x^3-(p+q+r)x^2+(pq+pr+qr)x-pqr$. Comparing with the standard form $x^3-a{x}^2+bx-c=0$:

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

The second term carries the sum of roots, the third term carries pairwise products, and the constant term is the product of all three roots. See vieta-formulas for the quadratic analogue.

Rational Root Theorem (§722)

Since $c=pqr$, every rational root must be a divisor of the constant term $c$. This sharply limits the candidates to try.

Procedure:

1. List all integer divisors of the constant term.
2. Substitute each in turn until one satisfies the equation.
3. Once a root $r$ is found, divide the cubic by $(x-r)$ to obtain a quadratic, then solve that.

Example. $x^3-x-6=0$: divisors of 6 are $\pm 1,2,3,6$. Trial gives $x=2$ (since $8-2-6=0$). Dividing by $(x-2)$ yields $x^2+2x+3=0$, with roots $x=-1\pm \sqrt{-2}$ (imaginary).

Handling Fractional Coefficients (§723–724)

Non-unit leading coefficient (§723). If the equation contains fractions, substitute $x=y/n$ for a suitable integer $n$ to obtain an integer-coefficient equation, then apply the rational root test.

Reciprocal substitution (§724). If the leading coefficient is not 1 but the constant term is 1, substituting $x=1/z$ reverses the equation (multiplying through by $z^3$), often producing a simpler set of trial divisors.

Sign Rule for Positive and Negative Roots (§725)

Euler states a version of Descartes’ rule of signs:

• Each sign change between consecutive terms corresponds to a positive root.
• Each run of same sign corresponds to a negative root.

This tells us whether to try positive or negative divisors of the constant term.

Example (§726). $x^3+x^2-34x+56=0$: signs are $+,+,-,+$—two changes, one run—so two positive roots and one negative root. The roots turn out to be $x=2$, $x=4$, and $x=-7$.

Worked Problems (§727–733)

§	Problem	Equation (simplified)	Answer
727	Two numbers, difference 12, product·sum = 14560	$x^3+18x^2+72x=7280$	14 and 26
728	Difference 18, sum × difference of cubes = 275184	$x^3+27x^2+270x=1576$	4 and 22
729	Difference 720, lesser × √greater = 20736	$u^2(u+5)=144$	576 and 1296
730	Same as 729, simpler substitution	$z^3-5z-12=0$	same
731	Difference 12, diff × sum-of-cubes = 102144	$y^3+9y^2+54y=424$	8 and 20
732	Partnership: each contributes $10x$, gain $(x+6)\%$, profit 392	$y^3+3y^2=490$	14 partners
733	Stock 8240, each contributes $40x$, gain $x\%$, remainder 224	$x^3-25x^2+206x-560=0$	7, 8, or 10 merchants (three valid solutions)

Problem 733 is notable: the equation has three positive real roots, all physically valid, giving three consistent answers to the word problem. Euler verifies all three in a table.

Related pages

• ch1.4.10-pure-cubic-equations
• ch1.4.12-cardanos-rule
• cubic-equations
• vieta-formulas
• rational-root-theorem
• quadratic-equations
• imaginary-numbers