Răspuns:
Explicație pas cu pas:
(√5+2)ˣ+(√5-2)ˣ=18
Observam ca (√5+2)(√5-2)=5-4=1
(√5+2)(√5-2)=1 de aici scot (√5+2)
[tex]\sqrt{5}+2=\frac{1}{\sqrt{5} -1}[/tex]
Notam (√5+2)ˣ=t
atunci (√5-2)ˣ=[tex]\frac{1}{t}[/tex]
vom avea
[tex]t+\frac{1}{t} =18\\[/tex] aducem la acelasi numitor comun "t" si eliminam numitorul
t²-18t+1=0
Δ=324-4=320
√Δ=8√5
[tex]t_{1}=\frac{18+8\sqrt{5} }{2} =9+4\sqrt{5} =(\sqrt{5} +2)^2\\\\t_{2}=\frac{18-8\sqrt{5} }{2} =9-4\sqrt{5} =(\sqrt{5}-2)^2[/tex]
t=(√5+2)²=(√5+2)ˣ⇒ x=2
t=(√5-2)²=(√5+2)ˣ=(√5-2)⁻x⇒ x=-2
