Question

Rezolvati in multimea R ecuatia : $\frac{8}{x-8} + \frac{10}{x-6} + \frac{12}{x-4} + \frac{14}{x-2} = 4.$

Answer 1

Se aduce toata ecuatia la acelasi numitor comun:$(x-8)(x-6)(x-4)(x-2).$ fara a mai trece numitorii. Se ajunge la ecuatia:
$-4x^4+124x^3-1240x^2+4480x-6400=0$
Cautam radacini printre divizorii lui 6400. Prin calcul se observa ca 5 si 16 sunt radacini.
Impartim polinomul $-4x^4+124x^3-1240x^2+4480x-6400$ la $-4(x-5)(x-16)$ si obtinem $x^2-10x+20$.
In concluzie:
[tex] \displaystyle -4(x-5)(x-16)(x^2-10x+20)=0\\ x-5=0=>x_1=5\\ x-16=0=>x_2=16\\ x^2-10x+20=0=> \left \{ {x_3=5+\sqrt{5}} \atop {x_4=5-\sqrt{5}}} \right. [/tex]