(√(5 + 2√6))^x + (√(5-2√6))^x =10
т.к
(√(5 + 2√6))·(√(5-2√6))=√(25-24)=1 ,
и [(√(5 + 2√6))^x ]·[ (√(5-2√6))^x]= [(√(5 + 2√6)) (√(5-2√6)]^x=1^x=1
тогда [(√(5 + 2√6))^x ]=1/[ (√(5-2√6))^x]
[(√(5 + 2√6))^x ]=t>0
(√(5 + 2√6))^x + (√(5-2√6))^x =10 ⇔ t+1/t=10 ⇒
t²-10t+1=0 ⇔ t1=5-2√6 t2=5-+2√6 ⇒
1) [(√(5 + 2√6))^x ]=(5-2√6) (5 + 2√6)^(x/2)= (5 + 2√6)^(-1) ⇒x/2=-1 x=-2
2) [(√(5 + 2√6))^x ]=(5+2√6) (5 + 2√6)^(x/2)= (5 + 2√6) ⇒x/2=1 x=2