Quadratic equation (original) (raw)
Στα μαθηματικά, δευτεροβάθμια εξίσωση ονομάζεται κάθε δευτέρου βαθμού. Μερικές φορές αναφέρεται και ως τετραγωνική εξίσωση. Η γενική μορφή μιας δευτεροβάθμιας εξίσωσης είναι: όπου τα γράμματα α, β και γ παριστάνουν σταθερούς αριθμούς, με Οι σταθερές α, β και γ ονομάζονται συντελεστές, με το α να είναι ο συντελεστής του x2, το β να είναι ο συντελεστής του x και γ ο σταθερός όρος. Οι συντελεστές μπορεί να είναι πραγματικοί ή μιγαδικοί αριθμοί.
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| dbo:abstract | Una equació de segon grau, anomenada també equació quadràtica, és una equació polinòmica on el grau més alt dels diversos monomis que la integren és 2. La seva expressió general és: on a ≠ 0. Les equacions de segon grau es resolen mitjançant la fórmula: , que proporciona les dues solucions complexes que té, d'acord amb el teorema fonamental de l'àlgebra. Per comprovar si aquestes solucions són també reals, es pot fer observant el discriminant de l'equació, que correspon al terme dins l'arrel quadrada: . Si: * Les dues solucions són reals. * L'equació té una sola solució real (doble), que ve donada per . * No existeixen solucions en els reals. (ca) في الرياضيات وبالتحديد في الجبر الابتدائي، المعادلة التربيعية (بالإنجليزية: Quadratic equation) هي معادلة جبرية أحادية المتغير من الدرجة الثانية، تكتب وفق الصيغة العامة حيث يمثل المجهول أو المتغير أما ، ، فيطلق عليها الثوابت أو المعاملات. يطلق على المعامل الرئيسي وعلى الحد الثابت . ويشترط أن يكون . أما إذا كان عندها تصبح المعادلة معادلة خطية لأن عنصر ال لم يعد موجوداً. يتم إيجاد حلول (أو جذور) المعادلة التربيعية باستعمال عدة طرق: باستعمال الصيغة التربيعية أو طريقة إكمال المربع أو طريقة حساب المميز أو طريقة الرسم البياني.تُسمى قيم المجهول x التي تحقق المعدالة حلا للمعادلة (أو حلحلةً لها)، أو جذورا لها أو أصفارا لها. للمعادلة التربيعية جذران على الأكثر. إذا وجد للمعادلة التربيعية جذرا واحدا فقط، فإنه يُقال عنه أنه جذر مزدوج. (ar) Jako kvadratická rovnice se v matematice označuje algebraická rovnice druhého stupně, tzn. rovnice o jedné neznámé, ve které neznámá vystupuje ve druhé mocnině (x²). V základním tvaru vypadá následovně: Zde jsou a, b, c nějaká čísla (obvykle reálná čísla, pro komplexní čísla vizte níže), tzv. koeficienty této rovnice, x je neznámá (obvykle se předpokládá reálná nebo komplexní). Koeficient a je vždy různý od nuly, neboť pro a = 0 se jedná o lineární rovnici. Často se kvadratická rovnice vyjadřuje v základním (normovaném) tvaru, kde a = 1. Do tohoto tvaru lze převést každou kvadratickou rovnici jejím vydělením koeficientem a. Jednotlivé mají také svá pojmenování: ax2 je kvadratický člen, bx je lineární člen a c absolutní člen. (cs) Στα μαθηματικά, δευτεροβάθμια εξίσωση ονομάζεται κάθε δευτέρου βαθμού. Μερικές φορές αναφέρεται και ως τετραγωνική εξίσωση. Η γενική μορφή μιας δευτεροβάθμιας εξίσωσης είναι: όπου τα γράμματα α, β και γ παριστάνουν σταθερούς αριθμούς, με Οι σταθερές α, β και γ ονομάζονται συντελεστές, με το α να είναι ο συντελεστής του x2, το β να είναι ο συντελεστής του x και γ ο σταθερός όρος. Οι συντελεστές μπορεί να είναι πραγματικοί ή μιγαδικοί αριθμοί. (el) Eine quadratische Gleichung ist eine Gleichung, die sich in der Form mit schreiben lässt. Hierbei sind Koeffizienten; ist die Unbekannte.Ist zusätzlich , spricht man von einer reinquadratischen Gleichung. Ihre Lösungen lassen sich anhand der Formel bestimmen. Im Bereich der reellen Zahlen kann die quadratische Gleichung keine, eine oder zwei Lösungen besitzen. Ist der Ausdruck unter der Wurzel negativ, so existiert keine Lösung; ist er Null, so existiert eine Lösung; wenn er positiv ist, so existieren zwei Lösungen. Die linke Seite dieser Gleichung ist der Term einer quadratischen Funktion (allgemeiner ausgedrückt: ein Polynom zweiten Grades), ; der Funktionsgraph dieser Funktion im kartesischen Koordinatensystem ist eine Parabel. Geometrisch beschreibt die quadratische Gleichung die Nullstellen dieser Parabel. (de) Una ecuación de segundo grado o ecuación cuadrática de una variable es aquella que tiene la expresión general: donde es la variable, y , y constantes; es el coeficiente cuadrático (distinto de cero), el coeficiente lineal y es el término independiente. Este polinomio se puede interpretar mediante la gráfica de una función cuadrática, es decir, por una parábola. Esta representación gráfica es útil, porque las abscisas de las intersecciones o punto de tangencia de esta gráfica, en el caso de existir, con el eje son las raíces reales de la ecuación. Si la parábola no corta el eje las raíces son números complejos. El primer caso (raíces reales) corresponde a un discriminante positivo, y el segundo (raíces complejas) a uno negativo. (es) Matematikan, aldagai bakarreko bigarren mailako ekuazioa edo ekuazio koadratikoa , era osoan, honela adierazten den aldagai bakarreko ekuazio polinomiko bat da: Ekuazioa ebaztean, ezezaguna den x aldagaiaren balioa zehaztea da helburua, hau da, ekuazioaren erroak edo soluzioak ateratzea, a, b eta c zenbakizko konstanteak izanik. Konstante hauei koefiziente deritze. Definizioz, bigarren mailako ekuazioan a ≠ 0 bete behar da, bestela lehenengo mailako ekuazio bat izango bailitzateke. a=1 betetzen denean, x2+bx+c=0 ekuazioetan alegia, ekuazio koadratikoa monikoa dela esaten da . Bigarren mailako ekuazio osatugabeak ere badaude , baina agertzen ez diren koefizienteak 0 bihurtuz aise aldatzen dira adierazpen orokorrera: Bigarren mailako ekuazioek aplikazio zabalak dituzte zientzian, hala-nola fisikan, azeleraziozko mugimenduen aztertzeko . (eu) San ailgéabar, is éard is cothromóid chearnach ann (ón Laidin quadratus 'cearnóg') ná aon chothromóid is féidir a atheagrú i bhfoirm chaighdeánach mar áit a seasann x d'uimhreacha anaithnid, agus a, b, agus c d'uimhreacha aitheanta, áit a seasann a ≠ 0 . Má tá a = 0, ansin tá an chothromóid líneach, ní cearnach, mar níl aon téarma ann. Is iad na huimhreacha a, b, agus c comhéifeachtaí na cothromóide agus is féidir iad a idirdhealú ach an chomhéifeacht chearnach, an chomhéifeacht líneach agus an téarma tairiseach nó saor a ghlaoch orthu, faoi seach. (ga) Persamaan kuadrat adalah suatu persamaanberorde dua. Bentuk umum dari persamaan kuadrat adalah dengan cara Huruf-huruf a, b dan c disebut sebagai koefisien: koefisien kuadrat a adalah koefisien dari , koefisien linier b adalah koefisien dari x, dan c adalah koefisien konstan atau disebut juga suku bebas. (in) In algebra, a quadratic equation (from Latin quadratus 'square') is any equation that can be rearranged in standard form as where x represents an unknown value, and a, b, and c represent known numbers. One supposes generally that a ≠ 0; those equations with a = 0 are considered degenerate because the equation then becomes linear or even simpler. The numbers a, b, and c are the coefficients of the equation and may be distinguished by calling them, respectively, the quadratic coefficient, the linear coefficient and the constant or free term. The values of x that satisfy the equation are called solutions of the equation, and roots or zeros of the expression on its left-hand side. A quadratic equation has at most two solutions. If there is only one solution, one says that it is a double root. If all the coefficients are real numbers, there are either two real solutions, or a single real double root, or two complex solutions that are complex conjugates of each other. A quadratic equation always has two roots, if complex roots are included; and a double root is counted for two. A quadratic equation can be factored into an equivalent equation where r and s are the solutions for x. The quadratic formula expresses the solutions in terms of a, b, and c. Completing the square is one of several ways for getting it. Solutions to problems that can be expressed in terms of quadratic equations were known as early as 2000 BC. Because the quadratic equation involves only one unknown, it is called "univariate". The quadratic equation contains only powers of x that are non-negative integers, and therefore it is a polynomial equation. In particular, it is a second-degree polynomial equation, since the greatest power is two. (en) 二次方程式（にじほうていしき、英: quadratic equation）とは、数学において、二次の多項式関数の零点集合を表す条件のことである。 その零点集合については、特に実数係数であるものについて、幾何学的考察が歴史的に行われ、よく知られている（二元二次方程式については円錐曲線を、一般の多変数二次方程式については二次曲面を参照するとよい）。 以下では、未知数が1個の場合を中心に取り扱う。二次方程式は次数が 2 の代数方程式のことであり、一般に未知数を x として の形で表される。二次方程式を解くには、二次方程式の解の公式が知られている他、平方完成を利用する方法、因数分解を利用する方法などがよく知られている。 一元二次方程式を解くことと同値である問題に対する解法は、紀元前20世紀ごろには既に知られていた。 (ja) En mathématiques, une équation du second degré, ou équation quadratique, est une équation polynomiale de degré 2, c'est-à-dire qu'elle peut s'écrire sous la forme : Dans cette équation, x est l'inconnue les lettres a, b et c représentent les coefficients, avec a différent de 0. a est le coefficient quadratique, b est le coefficient linéaire, et c est un terme constant. Dans l'ensemble des nombres réels, une telle équation admet au maximum deux solutions, qui correspondent aux abscisses des éventuels points d'intersection de la parabole d'équation y = ax2 + bx + c avec l'axe des abscisses dans le plan muni d'un repère cartésien. La position de cette parabole par rapport à l'axe des abscisses, et donc le nombre de solutions (0, 1 ou 2) est donnée par le signe du discriminant. Ce dernier permet également d'exprimer facilement les solutions, qui sont aussi les racines de la fonction du second degré associée. Sur le corps des nombres complexes, une équation du second degré a toujours exactement deux racines distinctes ou une racine double. Dans l'algèbre des quaternions, une équation du second degré peut avoir une infinité de solutions. (fr) In matematica, un'equazione di secondo grado o quadratica ad un'incognita è un'equazione algebrica in cui il grado massimo con cui compare l'incognita è 2, ed è sempre riconducibile alla forma: , dove sono numeri reali o complessi. Per il teorema fondamentale dell'algebra, le soluzioni (dette anche radici o zeri dell'equazione) delle equazioni di secondo grado nel campo complesso sono sempre due, se contate con la loro molteplicità. Nel campo reale invece le equazioni quadratiche possono ammettere due soluzioni, una soluzione doppia, oppure nessuna soluzione. Sono poi particolarmente semplici da risolvere le cosiddette equazioni incomplete, dove alcuni coefficienti sono uguali a zero. Il grafico della funzione nel piano cartesiano è una parabola, la cui concavità dipende dal segno di . Più precisamente: se la parabola ha la concavità rivolta verso l'alto, se la parabola ha la concavità rivolta verso il basso. (it) 이차방정식(二次方程式, 영어: quadratic equation)은 최고차항의 차수가 2인 다항 방정식이다. 에 관한 이차 방정식의 일반적인 형태는 와 같고, 여기서 는 변수, 와 는 각각 의 계수라고 하며, 는 상수항이라고 부른다. 일반적으로 인수분해를 이용해 풀이한다. 여기에서 에서 a의 값이 -이면 아래로 내려가고 +이면 위로 올라간다. 그리고 |a |
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| dbp:date | 2007-10-22 (xsd:date) 2007-11-10 (xsd:date) October 2017 (en) September 2021 (en) |
| dbp:id | p/q076050 (en) |
| dbp:postText | : this is linear, not quadratic (en) |
| dbp:reason | without indication on the numerical accuracy, the figure and its discussion are nonsensical. At least the difference with the exact value of the root must also appear. (en) |
| dbp:title | Quadratic equation (en) Quadratic equations (en) |
| dbp:url | https://web.archive.org/web/20071022022143/http:/plus.maths.org/issue30/features/quadratic/index-gifd.html https://web.archive.org/web/20071022022143/https:/web.archive.org/web/20220000000000*/http:/quadraticformulacalculator.org https://web.archive.org/web/20071110232247/http:/plus.maths.org/issue29/features/quadratic/index-gifd.html |
| dbp:urlname | QuadraticEquation (en) |
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| rdf:type | owl:Thing |
| rdfs:comment | Στα μαθηματικά, δευτεροβάθμια εξίσωση ονομάζεται κάθε δευτέρου βαθμού. Μερικές φορές αναφέρεται και ως τετραγωνική εξίσωση. Η γενική μορφή μιας δευτεροβάθμιας εξίσωσης είναι: όπου τα γράμματα α, β και γ παριστάνουν σταθερούς αριθμούς, με Οι σταθερές α, β και γ ονομάζονται συντελεστές, με το α να είναι ο συντελεστής του x2, το β να είναι ο συντελεστής του x και γ ο σταθερός όρος. Οι συντελεστές μπορεί να είναι πραγματικοί ή μιγαδικοί αριθμοί. (el) Una ecuación de segundo grado o ecuación cuadrática de una variable es aquella que tiene la expresión general: donde es la variable, y , y constantes; es el coeficiente cuadrático (distinto de cero), el coeficiente lineal y es el término independiente. Este polinomio se puede interpretar mediante la gráfica de una función cuadrática, es decir, por una parábola. Esta representación gráfica es útil, porque las abscisas de las intersecciones o punto de tangencia de esta gráfica, en el caso de existir, con el eje son las raíces reales de la ecuación. Si la parábola no corta el eje las raíces son números complejos. El primer caso (raíces reales) corresponde a un discriminante positivo, y el segundo (raíces complejas) a uno negativo. (es) San ailgéabar, is éard is cothromóid chearnach ann (ón Laidin quadratus 'cearnóg') ná aon chothromóid is féidir a atheagrú i bhfoirm chaighdeánach mar áit a seasann x d'uimhreacha anaithnid, agus a, b, agus c d'uimhreacha aitheanta, áit a seasann a ≠ 0 . Má tá a = 0, ansin tá an chothromóid líneach, ní cearnach, mar níl aon téarma ann. Is iad na huimhreacha a, b, agus c comhéifeachtaí na cothromóide agus is féidir iad a idirdhealú ach an chomhéifeacht chearnach, an chomhéifeacht líneach agus an téarma tairiseach nó saor a ghlaoch orthu, faoi seach. (ga) Persamaan kuadrat adalah suatu persamaanberorde dua. Bentuk umum dari persamaan kuadrat adalah dengan cara Huruf-huruf a, b dan c disebut sebagai koefisien: koefisien kuadrat a adalah koefisien dari , koefisien linier b adalah koefisien dari x, dan c adalah koefisien konstan atau disebut juga suku bebas. (in) 二次方程式（にじほうていしき、英: quadratic equation）とは、数学において、二次の多項式関数の零点集合を表す条件のことである。 その零点集合については、特に実数係数であるものについて、幾何学的考察が歴史的に行われ、よく知られている（二元二次方程式については円錐曲線を、一般の多変数二次方程式については二次曲面を参照するとよい）。 以下では、未知数が1個の場合を中心に取り扱う。二次方程式は次数が 2 の代数方程式のことであり、一般に未知数を x として の形で表される。二次方程式を解くには、二次方程式の解の公式が知られている他、平方完成を利用する方法、因数分解を利用する方法などがよく知られている。 一元二次方程式を解くことと同値である問題に対する解法は、紀元前20世紀ごろには既に知られていた。 (ja) 이차방정식(二次方程式, 영어: quadratic equation)은 최고차항의 차수가 2인 다항 방정식이다. 에 관한 이차 방정식의 일반적인 형태는 와 같고, 여기서 는 변수, 와 는 각각 의 계수라고 하며, 는 상수항이라고 부른다. 일반적으로 인수분해를 이용해 풀이한다. 여기에서 에서 a의 값이 -이면 아래로 내려가고 +이면 위로 올라간다. 그리고 |a |
| rdfs:label | معادلة تربيعية (ar) Equació de segon grau (ca) Kvadratická rovnice (cs) Quadratische Gleichung (de) Δευτεροβάθμια εξίσωση (el) Ecuación de segundo grado (es) Bigarren mailako ekuazio (eu) Cothromóid chearnach (ga) Persamaan kuadrat (in) Équation du second degré (fr) Equazione di secondo grado (it) 이차 방정식 (ko) 二次方程式 (ja) Vierkantsvergelijking (nl) Równanie kwadratowe (pl) Quadratic equation (en) Equação quadrática (pt) Квадратное уравнение (ru) 一元二次方程 (zh) Квадратне рівняння (uk) Andragradsekvation (sv) |
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