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Una equació de segon grau es pot escriure en la forma general

                               bold italic a bold italic x to the power of bold 2 bold plus bold italic b bold italic x bold plus bold italic c bold equals bold 0 bold space bold space bold space bold space bold space bold space space left parenthesis a not equal to 0 right parenthesis       
La resolem amb la fórmula: 

                        bold italic x bold equals fraction numerator bold minus bold b bold plus-or-minus square root of bold b to the power of bold 2 bold minus bold 4 bold times bold a bold times bold c end root over denominator bold 2 bold times bold a end fraction

El valor de dins l'arrel es diu discriminant i es denota Δ , així doncs bold increment bold equals bold b to the power of bold 2 bold minus bold 4 bold times bold a bold times bold c 

Segons el valor del discriminant podrem afirmar si l'equació té o no solució.

      • Si el discriminant és Δ>0 l'equació té dues solucions diferents
      • Si el discriminant és Δ=0 l'equació té una solució repetida
      • Si el discriminant és Δ<0 l'equació no té solució
- Si l'equació té els tres coeficients no nuls l'equació es diu completa i aplicarem la fórmula per resoldre.

 

Exemple 1

 bold 2 bold italic x to the power of bold 2 bold plus bold 5 bold italic x bold minus bold 1 bold equals bold 0 bold space bold space bold space space tenim bold space space bold space bold italic a bold equals bold 2 bold comma bold space bold space bold space bold space bold italic b bold equals bold 5 bold comma bold space bold space bold space bold italic c bold equals bold minus bold 1 bold space 


     bold italic x bold equals fraction numerator bold minus bold b bold plus-or-minus square root of bold b to the power of bold 2 bold minus bold 4 bold times bold a bold times bold c end root over denominator bold 2 bold times bold a end fraction bold equals fraction numerator bold minus bold 5 bold plus-or-minus square root of bold 5 to the power of bold 2 bold minus bold 4 bold times bold 2 bold times bold left parenthesis bold minus bold 1 bold right parenthesis end root over denominator bold 2 bold times bold 2 end fraction bold equals fraction numerator bold minus bold 5 bold plus-or-minus square root of bold 25 bold plus bold 8 end root over denominator bold 4 end fraction bold equals bold space bold space bold space fraction numerator bold minus bold 5 bold plus-or-minus square root of bold 33 over denominator bold 4 end fraction bold equals fraction numerator bold minus bold 5 bold plus-or-minus bold 5 bold comma bold 74 over denominator bold 4 end fraction bold equals open curly brackets table attributes columnalign left end attributes row cell bold x subscript bold 1 bold equals fraction numerator bold minus bold 5 bold minus bold 5 bold comma bold 74 over denominator bold 4 end fraction bold equals fraction numerator bold minus bold 10 bold comma bold 74 over denominator bold 4 end fraction bold equals bold minus bold 2 bold comma bold 685 end cell row cell bold x subscript bold 2 bold equals fraction numerator bold minus bold 5 bold plus bold 5 bold comma bold 74 over denominator bold 4 end fraction bold equals fraction numerator bold 0 bold comma bold 74 over denominator bold 4 end fraction bold equals bold 0 bold comma bold 185 end cell end table close

Exemple 2

 2 x squared plus 3 x minus 2 equals 0 space space space space space space space space space space x equals fraction numerator negative 3 plus-or-minus square root of left parenthesis negative 3 right parenthesis squared minus 4 times 2 times left parenthesis negative 2 right parenthesis end root over denominator 2 times 2 end fraction equals fraction numerator negative 3 plus-or-minus square root of 9 plus 16 end root over denominator 4 end fraction equals fraction numerator negative 3 plus-or-minus square root of 25 over denominator 4 end fraction equals fraction numerator negative 3 plus-or-minus 5 over denominator 4 end fraction equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator negative 3 plus 5 over denominator 4 end fraction equals 2 over 4 equals 1 half end cell row cell fraction numerator negative 3 minus 5 over denominator 4 end fraction equals fraction numerator negative 8 over denominator 4 end fraction equals negative 2 end cell end table close

Exemple 3

 2 x squared minus 3 x minus 2 equals 0 space space space space space space space space space space x equals fraction numerator negative left parenthesis negative 3 right parenthesis plus-or-minus square root of left parenthesis negative 3 right parenthesis squared minus 4 times 2 times left parenthesis negative 2 right parenthesis end root over denominator 2 times 2 end fraction equals fraction numerator 3 plus-or-minus square root of 9 plus 16 end root over denominator 4 end fraction equals fraction numerator 3 plus-or-minus square root of 25 over denominator 4 end fraction equals fraction numerator 3 plus-or-minus 5 over denominator 4 end fraction equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator 3 plus 5 over denominator 4 end fraction equals 8 over 4 equals 2 end cell row cell fraction numerator 3 minus 5 over denominator 4 end fraction equals fraction numerator negative 2 over denominator 4 end fraction equals negative 1 half end cell end table close

Exemple 4

 x squared minus 5 x plus 6 equals 0 space space space space space space space space space space x equals fraction numerator negative left parenthesis negative 5 right parenthesis plus-or-minus square root of left parenthesis negative 5 right parenthesis squared minus 4 times 1 times 6 end root over denominator 2 times 1 end fraction equals fraction numerator 5 plus-or-minus square root of 25 minus 24 end root over denominator 2 end fraction equals fraction numerator 5 plus-or-minus square root of 1 over denominator 2 end fraction equals fraction numerator 5 plus-or-minus 1 over denominator 2 end fraction equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator 5 plus 1 over denominator 2 end fraction equals 6 over 2 equals 3 end cell row cell fraction numerator 5 minus 1 over denominator 2 end fraction equals 4 over 2 equals 2 end cell end table close

Exemple 5

x squared minus 5 x plus 7 equals 0 space space space space space space space space space space x equals fraction numerator negative left parenthesis negative 5 right parenthesis plus-or-minus square root of left parenthesis negative 5 right parenthesis squared minus 4 times 1 times 7 end root over denominator 2 times 1 end fraction equals fraction numerator 5 plus-or-minus square root of 25 minus 28 end root over denominator 2 end fraction equals fraction numerator 5 plus-or-minus square root of negative 3 end root over denominator 2 end fraction space space space space N o space t é space s o l u c i ó

      Aquesta equació no té solució en els nombres reals perquè no existeix l'arrel quadrada d'un negatiu. 

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