@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-V91WMW66-Q
  skos:prefLabel "équation polynomiale"@fr, "polynomial equation"@en ;
  a skos:Concept ;
  skos:narrower psr:-LK8XNN3X-R .

psr:-MZK5LLKD-P
  skos:prefLabel "completing the square"@en, "complétion du carré"@fr ;
  a skos:Concept ;
  skos:broader psr:-LK8XNN3X-R .

psr:-LK8XNN3X-R
  skos:prefLabel "quadratic equation"@en, "équation du second degré"@fr ;
  skos:inScheme psr: ;
  a skos:Concept ;
  skos:definition """En mathématiques, une <b>équation du second degré</b>, ou <b>équation quadratique</b>, est une équation polynomiale de degré 2, c'est-à-dire qu'elle peut s'écrire sous la forme :  <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle ax^{2}+bx+c=0}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>a</mi>         <msup>           <mi>x</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo>+</mo>         <mi>b</mi>         <mi>x</mi>         <mo>+</mo>         <mi>c</mi>         <mo>=</mo>         <mn>0</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle ax^{2}+bx+c=0}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23e70cfa003f402d108ec04d97983fb62f69536e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.89ex; height:2.843ex;" alt="{\\\\displaystyle ax^{2}+bx+c=0}"></span></center> Dans cette équation, <span class="texhtml mvar" style="font-style:italic;">x</span> est l'inconnue les lettres <i><span class="texhtml">a</span></i>, <i><span class="texhtml">b</span></i> et <i><span class="texhtml">c</span></i> représentent les coefficients, avec <span class="texhtml"><i>a</i></span> différent de 0. <i><span class="texhtml">a</span></i> est le coefficient quadratique, <i><span class="texhtml">b</span></i> est le coefficient linéaire, et <i><span class="texhtml">c</span></i> est un terme constant où le polynome est défini sur  <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\mathbb {R} }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mi mathvariant="double-struck">R</mi>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbb {R} }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\\\\displaystyle \\\\mathbb {R} }"></span>. 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 <span class="nowrap"><span class="texhtml"><i>y = ax</i><sup>2</sup> + <i>bx</i> + <i>c</i></span></span> 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.  
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/%C3%89quation_du_second_degr%C3%A9">https://fr.wikipedia.org/wiki/%C3%89quation_du_second_degr%C3%A9</a>)"""@fr, """In algebra, a <b>quadratic equation</b> (from Latin <i>quadratus</i> 'square') is any equation that can be rearranged in standard form as <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle ax^{2}+bx+c=0\\\\,,}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>a</mi>         <msup>           <mi>x</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo>+</mo>         <mi>b</mi>         <mi>x</mi>         <mo>+</mo>         <mi>c</mi>         <mo>=</mo>         <mn>0</mn>         <mspace width="thinmathspace"></mspace>         <mo>,</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle ax^{2}+bx+c=0\\\\,,}</annotation>   </semantics> </math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d40196d521aae8b791055b7da8f8844357969a1f" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.923ex; height:3.009ex;" alt="{\\\\displaystyle ax^{2}+bx+c=0\\\\,,}"></div> where <span class="texhtml"><i>x</i></span> represents an unknown value, and <span class="texhtml"><i>a</i></span>, <span class="texhtml"><i>b</i></span>, and <span class="texhtml"><i>c</i></span> represent known numbers, where <span class="texhtml"><i>a</i> ≠ 0</span>. (If <span class="texhtml"><i>a</i> = 0</span> and <span class="texhtml"><i>b</i> ≠ 0</span> then the equation is linear, not quadratic.) The numbers <span class="texhtml"><i>a</i></span>, <span class="texhtml"><i>b</i></span>, and <span class="texhtml"><i>c</i></span> are the <i>coefficients</i> of the equation and may be distinguished by respectively calling them, the <i>quadratic coefficient</i>, the <i>linear coefficient</i> and the <i>constant coefficient</i> or <i>free term</i>. The values of <span class="texhtml mvar" style="font-style:italic;">x</span> that satisfy the equation are called <i>solutions</i> of the equation, and <i>roots</i> or <i>zeros</i> 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  <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle ax^{2}+bx+c=a(x-r)(x-s)=0}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>a</mi>         <msup>           <mi>x</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo>+</mo>         <mi>b</mi>         <mi>x</mi>         <mo>+</mo>         <mi>c</mi>         <mo>=</mo>         <mi>a</mi>         <mo stretchy="false">(</mo>         <mi>x</mi>         <mo>−<!-- − --></mo>         <mi>r</mi>         <mo stretchy="false">)</mo>         <mo stretchy="false">(</mo>         <mi>x</mi>         <mo>−<!-- − --></mo>         <mi>s</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mn>0</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle ax^{2}+bx+c=a(x-r)(x-s)=0}</annotation>   </semantics> </math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d8eec520fbc9e80c17fa4795fad8f88218720b5" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.316ex; height:3.176ex;" alt="{\\\\displaystyle ax^{2}+bx+c=a(x-r)(x-s)=0}"></div> where <span class="texhtml mvar" style="font-style:italic;">r</span> and <span class="texhtml mvar" style="font-style:italic;">s</span> are the solutions for <span class="texhtml mvar" style="font-style:italic;">x</span>.  The quadratic formula <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle x={\\rac {-b\\\\pm {\\\\sqrt {b^{2}-4ac}}}{2a}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>x</mi>         <mo>=</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <mo>−<!-- − --></mo>               <mi>b</mi>               <mo>±<!-- ± --></mo>               <mrow class="MJX-TeXAtom-ORD">                 <msqrt>                   <msup>                     <mi>b</mi>                     <mrow class="MJX-TeXAtom-ORD">                       <mn>2</mn>                     </mrow>                   </msup>                   <mo>−<!-- − --></mo>                   <mn>4</mn>                   <mi>a</mi>                   <mi>c</mi>                 </msqrt>               </mrow>             </mrow>             <mrow>               <mn>2</mn>               <mi>a</mi>             </mrow>           </mfrac>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle x={\\rac {-b\\\\pm {\\\\sqrt {b^{2}-4ac}}}{2a}}}</annotation>   </semantics> </math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00c22777378f9c594c71158fea8946f2495f2a28" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.525ex; height:6.176ex;" alt="{\\\\displaystyle x={\\rac {-b\\\\pm {\\\\sqrt {b^{2}-4ac}}}{2a}}}"></div> expresses the solutions in terms of <span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>, and <span class="texhtml mvar" style="font-style:italic;">c</span>. Completing the square is one of several ways for deriving the formula. 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 <span class="texhtml"><i>x</i></span> 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. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Quadratic_equation">https://en.wikipedia.org/wiki/Quadratic_equation</a>)"""@en ;
  skos:broader psr:-V91WMW66-Q, psr:-W96QGKZX-0 ;
  dc:created "2023-08-04"^^xsd:date ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/%C3%89quation_du_second_degr%C3%A9>, <https://en.wikipedia.org/wiki/Quadratic_equation> ;
  skos:narrower psr:-MZK5LLKD-P, psr:-PD3N9LFZ-B ;
  skos:altLabel "équation quadratique"@fr ;
  skos:related psr:-SDKXNB3P-N ;
  dc:modified "2024-10-18"^^xsd:date .

psr: a skos:ConceptScheme .
psr:-SDKXNB3P-N
  skos:prefLabel "irrationnel quadratique"@fr, "quadratic irrational"@en ;
  a skos:Concept ;
  skos:related psr:-LK8XNN3X-R .

psr:-W96QGKZX-0
  skos:prefLabel "elementary algebra"@en, "algèbre élémentaire"@fr ;
  a skos:Concept ;
  skos:narrower psr:-LK8XNN3X-R .

psr:-PD3N9LFZ-B
  skos:prefLabel "Carlyle circle"@en, "cercle de Carlyle"@fr ;
  a skos:Concept ;
  skos:broader psr:-LK8XNN3X-R .