Fontem paginae Aequatio inspicere

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[[Fasciculus:Quadratic equation coefficients.png|thumb|Aequatio in tabula representari potest.  Hae tabulae monstrant aequationem quadraticam <math>a x^2 + b x + c = 0</math>, ubi quantitates a, b, c valores alios habent secundum colores:  in prima imagine, a = 0 et b = 0;  in seconda, a = 0 et c = 0;  in tertia, b = 0 et c = 0.  Figurae sunt [[parabola]]e.]]
:''Haec est de aequationibus mathematicis. Si de [[chemica]] quaeres, vide [[Aequatio chemica|aequationem chemicam]].''

'''Aequatio'''<ref>[http://books.google.com/books?id=e9bPtrEfYbIC&pg=PA206&lpg=PA206&dq=aequatio+differentialis&source=bl&ots=FtE6Y9r9uB&sig=ZnxaJhIn4EtlmY2nkucAX0Dv2Wg&hl=en&ei=QxTcTbz7II_6sgbatf3kDg&sa=X&oi=book_result&ct=result&resnum=2&ved=0CB4Q6AEwAQ#v=onepage&q=aequatio%20differentialis&f=false Die Streitschriften By Jakob Bernoulli, Jean Bernoulli, Herman Heine Goldstine, P. Radelet-de Grave] {{ling|Anglice|Latine}}</ref> est [[sententia]] [[mathematica]] quae dicit duas [[quantitas|quantitates]] esse aequas. Omnes aequationes habent symbolum aequalitatis "=", sicut

:<math>2 + 3 = 5</math>,

quod verbis dicitur "[[duo]] et [[tres]] sunt/aequant [[quinque]]," et 

:<math>x - x = 0</math>.

Si sententia dicit quantitatem aliquem maiorem vel minorem esse alia quantitate, est [[inaequatio]], sicut

:<math>2 + 4 > 5</math> (hoc est, "[[duo]] et [[quattuor]] sunt maiores quam [[quinque]]").

== Aequationes versus identitates ==
Aequationes supra sunt [[identitas mathematica|identitates]] mathematicae, aequationes quae sunt vera valoribus suarum partium neglectis. Aequatio cui non est identitas est:

:<math>x + 1 = 2</math>,

quae quippe est falsa infinito numero valoris dato, et vera est solum si <math>x = 1</math>. ''[[Solutio aequationis|Aequationem solvere]]'' significat invenire unicum valorem vel valores quibus aequatio vera est.

Saepius litterae primae [[abecedarium|abecedarii]], ut ''a, b, c,'' quantitates constantes nominant, et litterae ultimae, ut ''x, y, z,'' nominant quantitates variabiles quarum valores inveniendi.  Exempli gratia,

:<math>x + a = b</math> significat <math>x = b - a</math>,

in qua sententia scimus ''a'' et ''b'' et debemus solvere ut ''x'' sciamus.

== Proprietates ==
Si aequatio noscitur vera esse, in eam operationes hae sequentes faciantur et veram etiam habeas:

# Ullum valorem [[additio|adde]] ambobus lateribus:  si a = b, tum a + c = b + c.
# Ullum valorem [[subtractio|subtrahe]] de ambobus lateribus:  si a = b, tum a - c = b - c.
# Ambo latera [[multiplicatio|multiplicentur]] ullo valore:  si a = b, tum ac = bc.
# Ambo latera [[divisio|dividantur]] ullo valore excluso [[zero]]:  si a = b, et c ≠ 0, tum a/c = b/c.
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The algebraic properties (1-4) imply that equality is a [[congruence relation]] for a [[field (mathematics)|field]]; in fact, it is essentially the only one.

The most well known system of numbers which allows all of these operations is the [[real numbers]], which is an example of a [[field (mathematics)|field]].  However, if the equation were based on the [[natural number]]s for example, some of these operations (like division and subtraction) may not be valid as negative numbers and non-[[whole numbers]] are not allowed.  The [[integers]] are an example of an [[integral domain]] which does not allow all divisions as, again, whole numbers are needed.  However, subtraction is allowed, and is the [[inverse operator]] in that system.

If a function that is not [[injective]] is applied to both sides of a true equation, then the resulting equation will still be true, but it may be less useful. Formally, one has an [[Logical conditional|implication]], not an [[Logical biconditional|equivalence]], so the solution set may get larger. The functions implied in properties (1), (2), and (4) are always injective, as is (3) if we do not multiply by [[0 (number)|zero]]. Some generalized [[Product (mathematics)|products]], such as a [[dot product]], are never injective.
-->

== Aequationis gradus et canonica forma ==
Canonica aequationis cuiusdam forma appellatur forma quae formam "[[polynomium]] = [[zero]]" habeat.
Exempli gratia:
:x+2=0 (primus gradus)
:5x<sup>2</sup> - 9x + 17 = 0 (secundus gradus)

Gradus polynomii est denique etiam gradus aequationis.

== Quomodo possumus solvere aequationes gradus secundi ==
Aequationes gradus secundi (aut [[aequatio quadratica|aequationes quadraticae]] nominatae), quae habent formam 

<math> a x^2 + bx + c = 0 </math>,

habent solutionem

<math>
x = \frac {-b \pm \sqrt{b^2-4ac}}{2a}
</math>
             
ubi

# si <math>\Delta = b^2-4ac >0</math>, aequatio duas solutiones habet ambas reales,
# si <math>\Delta=0 </math>, aequatio tantum unam habet solutionem (aut melius, duas habet quae sunt concurrentes), denique, 
# si <math>\Delta<0</math>, aequatio duas distinctas solutiones complexas (partim realem et partim imaginariam) habet.

<math>\Delta = b^2 - 4ac</math> dicitur ''discriminans'' formae.

== Quomodo possumus solvere aequationes gradus terti ==

Ab-initio, consideratur aequationes gradus terti, quae habent formam 

<math> y^3 + qy + p = 0 </math>,

Eae habent solutionem

<math>u=\sqrt[3]{-\frac{q}{2}+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}};\quad v=\sqrt[3]{-\frac{q}{2}-\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}.</math>

ubi Delta est

<math>\Delta =\frac{q^2}{4}+\frac{p^3}{27} </math>, positivus vel negativus est.

* Si <math>\Delta</math>  major est  0, calcolantur duos numeros reales  <math>u</math> e <math>v</math>  qui  equali  sunt sic:
::<math>u=\sqrt[3]{-\frac{q}{2}+\sqrt{\Delta}}; \quad v=\sqrt[3]{-\frac{q}{2}-\sqrt{\Delta}},</math>

e  solutiones equationes sunt:
::<math>y_1=u+v;</math>
::<math>y_2=u\left(-\frac{1}{2}+i\frac{\sqrt{3}}{2}\right)+v\left(-\frac{1}{2}-i\frac{\sqrt{3}}{2}\right);</math>
::<math>y_3=u\left(-\frac{1}{2}-i\frac{\sqrt{3}}{2}\right)+v\left(-\frac{1}{2}+i\frac{\sqrt{3}}{2}\right).</math>

* Se <math>\Delta</math> minor ab 0 necesse est numerus complexus convertire 
::<math>-\frac{q}{2}+i\sqrt{-\Delta}</math>

{{NexInt}}
* [[Aequatio differentialis]] 
* [[Aequatio massae et energiae]]
* [[Diophantus]]
* [[Inaequatio]]
* [[Perfectio quadri]]
* [[Solutio aequationis]]
* [[Systema aequationum]]
*[[Hieronymus_Cardanus]]
*[[Quantitas_imaginaria]]

== Nota ==
<references />

== Nexus externi ==
* [http://www.wessa.net/math.wasp Machina quae aequationes et integralia describit]
* [http://eqworld.ipmnet.ru/ EqWorld] — sapientia de multis speciebus aequationum.
* [https://web.archive.org/web/20071028140036/http://www.numberz.co.uk/ES.html EquationSolver]
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{{Myrias|Mathematica}}

[[Categoria:Algebra]]
[[Categoria:Mathematica elementaria]]