Question

Cum se rezolva acest exercițiu?

Answer 1

Observam ca:
(√5+2)(√5-2)=√5²-4=5-4=1
Daca (√5+2)(√5-2)=1, inseamna ca √5-2=1/√5+2
Ecuatia devine:
(√5+2)^x+1/(√5+2)^x=18
Notam (√5+2)^x=t
t+1/t=18
Inmultim ecuatia cu t si avem:
t^2+1=18t
t^2-18t+1=0
Δ=18*18-4*1=324-4=320
√Δ=3√5
t1=(18+3√5)/2
t2=(18-3√5)/2
Cazul 1: (√5+2)^x=(18+3√5)/2
x=log in baza √5+2 din (18+3√5)/2
Cazul 2: (√5+2)^x=(18-3√5)/2
x=log in baza √5+2 din (18-3√5)/2