Equations

Summary: An equation is a formal statement of equality between two algebraic expressions; the central object of Section IV of Elements of Algebra, where Euler classifies equations by degree and establishes the transformation rules that preserve equality.

Sources: chapter-1.4.1, chapter-1.4.2, chapter-1.4.10, chapter-1.4.11, chapter-1.4.12, chapter-1.4.13, chapter-1.4.14, chapter-1.4.15, chapter-1.4.16

Last updated: 2026-05-03 (updated)

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Definition

An equation consists of two parts — the left-hand side and the right-hand side — separated by the sign $=$, asserting that the two expressions have the same value. (source: chapter-1.4.1, §571)

The standard goal is to deduce the value of an unknown quantity $x$ by systematically reducing the equation to the form $x=\text{known number}$.

Setting up equations from word problems

Euler’s method (§566–568):

1. Represent the unknown by a letter such as $x$.
2. Express every quantity mentioned in the problem in terms of $x$ and the given data.
3. Identify the condition the problem asserts, and write it as an equality of two expressions.
4. Solve the resulting equation.

The method generalises: for $n$ unknowns, one needs $n$ independent equations (§570).

Transformation rules

Two equal quantities remain equal under (§571):

• Adding or subtracting the same quantity on both sides
• Multiplying or dividing both sides by the same non-zero number
• Raising both sides to the same power
• Extracting the same root from both sides
• Taking logarithms of both sides

These are the only operations used throughout ch1.4.2-resolution-simple-equations.

Classification by degree (§572)

Degree	Defining property	Example
First (simple)	$x$ appears to power 1 only	$3x+7=22$
Second	$x^2$ appears after reduction	$x^2-5x+6=0$
Third	$x^3$ appears after reduction	$x^3=8$
Fourth	$x^4$ appears after reduction	$x^4-25x^2+60x-36=0$

Euler treats all four degrees in Section IV; the first degree is the subject of ch1.4.2-resolution-simple-equations and ch1.4.3-solution-of-questions.

Quadratic equations (second degree)

A quadratic always has exactly two solutions (roots), arising from the factored form $(x-p)(x-q)=0$. Euler further distinguishes:

• Pure quadratics ($a{x}^2+c=0$, no linear term): solved by $x=\pm \sqrt{-c/a}$. See ch1.4.5-pure-quadratic-equations.
• Mixt (complete) quadratics ($a{x}^2+bx+c=0$): solved by completing-the-square, yielding the quadratic formula. See ch1.4.6-mixt-quadratic-equations.

The sum and product of the two roots are directly readable from the coefficients (vieta-formulas). The sign of the discriminant ($a^2-4b$ for $x^2-ax+b=0$) determines whether the roots are real or imaginary.

Cubic equations (third degree)

A cubic always has exactly three roots, at least one of which is real (§706–713 and §719). Euler distinguishes:

• Pure cubics ($x^3=a$, no $x^2$ or $x$ term): solved by cube-root extraction; the two non-real roots involve $\sqrt{-3}$. See ch1.4.10-pure-cubic-equations.
• Complete cubics ($a{x}^3\pm b{x}^2\pm cx\pm d=0$): solved by the rational root test (rational roots must divide the constant term) followed by reduction to a quadratic. See ch1.4.11-complete-cubic-equations.
• General case when no rational root exists: use Cardan’s Rule (substitution to remove $x^2$, then Cardan’s formula). See ch1.4.12-cardanos-rule and cardanos-rule.

Vieta’s relations extend to cubics: for $x^3-a{x}^2+bx-c=0$ with roots $p,q,r$: $a=p+q+r$, $b=pq+pr+qr$, $c=pqr$ (vieta-formulas). See cubic-equations for a consolidated overview.

Quartic equations (fourth degree)

A quartic always has exactly four roots. Euler distinguishes pure quartics ($x^4=f$), incomplete quartics ($x^4+f{x}^2+g=0$), palindromic cases, and the general form. All four roots may be determined by:

• Rational roots: any rational root divides the constant term (rational-root-theorem); descartes-sign-rule narrows which signs to try.
• Bombelli’s method: reduce to an auxiliary cubic (the “Bombelli cubic”) via a difference-of-squares decomposition (bombelli-rule, ch1.4.14-bombelli-rule).
• Euler’s radical ansatz: assume $x=\sqrt{p}+\sqrt{q}+\sqrt{r}$ where $p,q,r$ are roots of an auxiliary cubic; four roots follow from sign combinations (ch1.4.15-new-method-quartic).

The full framework is at quartic-equations.

Equations of degree 5 and above

Euler notes (§780) that no general algebraic formula is known for degree 5 or higher. For these equations, and also for irrational roots of lower degrees, numerical approximation-methods must be used: iterative Newton-like correction or recurrence-series ratios.

The identical equation

An equation satisfied by every value of $x$ is called an identical equation (§597). It arises when both sides reduce to the same expression, and signals that the unknown is indeterminate — the problem has infinitely many solutions because the stated conditions are not independent. See the example in ch1.4.3-solution-of-questions.

Multiple unknowns

When two or more unknowns appear, a determinate problem provides exactly as many equations as unknowns. These simultaneous equations are treated in systems-of-linear-equations and ch1.4.4-resolution-two-or-more-equations.

Related pages

• ch1.4.1-solution-of-problems-in-general
• ch1.4.2-resolution-simple-equations
• ch1.4.3-solution-of-questions
• ch1.4.4-resolution-two-or-more-equations
• ch1.4.5-pure-quadratic-equations
• ch1.4.6-mixt-quadratic-equations
• ch1.4.9-nature-quadratic-equations
• quadratic-equations
• cubic-equations
• completing-the-square
• vieta-formulas
• discriminant
• rational-root-theorem
• cardanos-rule
• systems-of-linear-equations
• quartic-equations
• bombelli-rule
• descartes-sign-rule
• approximation-methods
• ch1.4.13-quartic-equations
• ch1.4.14-bombelli-rule
• ch1.4.15-new-method-quartic
• ch1.4.16-approximation
• logarithms
• powers-and-exponents