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Formula de Bhaskara - Problemas y ejercicios de matematica

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 Formula de Bhaskara

 

Deducción de la fórmula de Bhaskara para calcular las raíces de un polinomio de segundo grado usando un cambio de variable:

 

Dada la ecuación:

 ax^2+bx+c=0 \,   Buscamos calcular las raíces (valores de x que son soluciones de la ecuación)

 

Para ello comenzaremos a despejar la x, comenzaremos por dividir la ecucación entre el coeficiente principal a

obteniendo  

o sea 

 se puede simplificar si aplicamos el cambio de variable  2m = b/a y  n = c/a. Así la ecuación queda:

  • Aplicamos el cambio de variable
 x^2 + 2mx + n = 0 \,
  • Sumamos  m^2 para ajustar cuadrados, y pasamos n al otro lado
 x^2 + 2mx + m^2 = m^2 -n \,
  • Y lo contraemos
 (x+m)^2= m^2 - n \,
  • Aplicamos la raíz cuadrada a ambos lados
 x+m = \pm \sqrt{m^2-n} \,
  • Pasasmos restando m
 x = -m \pm \sqrt{m^2-n} \,
  • Deshaciendo la sustitución, m = b/2a y n = c/a
 x = -\frac{b}{2a} \pm \sqrt{\left(\frac{b}{2a}\right)^2- \frac{c}{a}} \,
  • Y operando llegamos a la siguiente ecuación:
 x = \frac{ -b \pm \sqrt{b^2-4ac}}{2a} \,

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