(5+2(6)^(1/2) )^(x^2-3)-(5-2(6)^(1/2) )^(x^2-3)=10 Solve for x

Genius Dude

Deepak Aggarwal

( 5 + 2(6)^1/2 ) ( 5 - 2(6)^1/2 ) = 25 - 24 = 1

So, ( 5 + 2(6)^1/2 ) = 1 / ( 5 - 2(6)^1/2 )

According to equation now,

Let ( 5 + 2(6)^1/2 )^{(x^2 - 3)} = y

So, y - (1/y) = 10

or, y² - 10y - 1 = 0

So, y = (10 + (104)^1/2) / 2 = 5 + (26)^1/2

So, x² - 3 = log (5 + (26)^1/2) / log (5 + 2(6)^1/2)

Therefore, x = [ { log (5 + (26)^1/2) / log (5 + 2(6)^1/2) } + 3 ]^1/2 (Ans)...........