Răspuns
1 , -1
Explicație pas cu pas:
Mai intai hai sa observam ca :
[tex](4+\sqrt{15})^x = \dfrac{1}{(4-\sqrt{15})^x}[/tex]
Prin urmare:
[tex](4+\sqrt{15})^x+ \dfrac{1}{(4+\sqrt{15})^x}=8[/tex]
Facem notatia : [tex](4+\sqrt{15})^x=t , t>0[/tex]
Ecuatia devine:
[tex]t+\dfrac{1}{t}=8 |\cdot t\\t^2+1=8t\\t^2-8t+1=0\\\Delta= 64-4= 60\Rightarrow \sqrt{\Delta}=2\sqrt{15}\\t_1= \dfrac{8+2\sqrt{15}}{2}=4+\sqrt{15}\\t_2=\dfrac{8-2\sqrt{15}}{2}=4-\sqrt{15}[/tex]
Se observa ca amandoua solutii sunt convenabile.
[tex](4+\sqrt{15})^x=4+\sqrt{15} \Rightarrow x=1\\(4+\sqrt{15})^x=4-\sqrt{15}\Rightarrow x=-1\\S:x\in \{\pm 1\}[/tex]
