根号2加1和的x次方加上根号2减1差的x次方等于6,求x

Question

根号2加1和的x次方加上根号2减1差的x次方等于6,求x

Answer 1

(√2+1)^x+(√2-1)^x=6
因为√2-1=1/(√2+1)
所以令a=(√2+1)^x,则(√2-1)^x=1/a
所以a+1/a=6
a^2-6a+1=0
a=3±2√2=(√2±1)^2
所以(√2+1)^x=(√2+1)^2,或(√2+1)^x=(√2-1)^2=1/(√2+1)^2=(√2+1)^(-2)
所以x=2,x=-2