sqrt(x+3)=8/sqrt(x-9)

√(x+3)=8/√(x-9)     multiply both sides by  √(x-9)

√(x-9) * √(x + 3)  = 8

√[(x - 9)*(x + 3) ] = 8     square both sides

(x - 9) (x + 3) = 64 

X^2 - 6x - 27 = 64

x^2 - 6x - 91 = 0       factor

(x - 13) ( x + 7) = 0      setting each factor to 0 we have    x = 13   or  x = -7    { reject -7 }

So

x = 13