Ecuaţia de gradul 2

ax² + bx + c =0, unde a,b,c ≠ 0 si a, b, c € R

după normare aⁱ = – b/a , bⁱ = c/a                                                       (1)

x² – aⁱx + bⁱ = 0                                                                                 (1’)

fie X1 si X2 soluţiile  ecuaţiei de gradul 2

(x – X1)(x – X2) = 0 =>

x² – (X1 + X2)x +X1X2 = 0                                                              (2)

din (1’) si (2) rezulta că soluţiile ecuaţiei trebuie să satisfacă următoarea egalitate:

X1 + X2 =  aⁱ                                                                                     (3)

X1X2 = bⁱ

Putem alege în mod convenabil  X1 şi X2 astfel încât:

X1 = aⁱ/2 + D                                                                                     (4) 

X2 = aⁱ/2 – D

(unde D € R)

Din (3) si (4) =>

X1 + X2 = (aⁱ/2 + D) + (aⁱ/2 – D) = aⁱ                                               (5)

=>

X1X2 = (aⁱ/2 + D) (aⁱ/2 – D) = aⁱ²/4 – D² = bⁱ                                   (6)

Din (6) putem obţine valoarea lui D:

D = rad (aⁱ²/4 – bⁱ)                                                                              (7)

Înlocuind pe D în (4) =>

X1 = aⁱ/2 + rad (aⁱ²/4 – bⁱ)                                                                   (4’)                                                                                                                                                                 

X2 = aⁱ/2 – rad (aⁱ²/4 – bⁱ)

(4’) + (1’) =>

X1 = (– b + rad (b² – 4ac))/2a                                                             (4’)                                                                                                                                                                

X2 = (– b – rad (b² – 4ac))/2a