(5-2*(6)^(1/2))^x+(5+2*(6)^(1/2))^x=98

Answer 1

my nomination of "trickiest" math question of the week.

(5 - 2√6)^x + (5 + 2√6)^x = 98

let A = 5-2√6 and B = 5+2√6
then AB = (5-2√6)(5+2√6) = 25-24 = 1
That's nice, so
AB=1 or B = 1/A

so we have
A^x + B^x = 98 or
A^x + (1/A)^x = 98

let A^x = y

y + 1/y = 98
y^2 + 1 = 98y
y^2 - 98y = -1, now complete the square
y^2 - 98y + 2401 = -1 + 2401
(y-49)^2 = 2400
y = 49 ± √2400

two cases:

1. y = 49-√2400
A^x = 49-√2400
(5-2√6)^x = 49-√2400
x(ln(5-2√6)) = ln(49-√2400)
x = 2

2. y = 49+√2400
(5-2√6)^x = 49+√2400
x(ln(5-2√6) = ln(49+√2400)
x = -2

both work, I checked on my calculator.

I hope somebody can find a better way.