Find the largest possible value of $x$ in the simplified form $x=\frac{a+b\sqrt{c}}{d}$ if $\frac{5x}{6}+1=\frac{3}{x}$, where $a,b,c,$ and $d$ are integers. What is $\frac{acd}{b}$?
Please help me convert this to proper latex, also, include an explanation please.
+128448 (5/6)x + 1 = 3/x multiply through by x
(5/6)x^2 + x = 3 rearrange as
(5/6)x^2 + x - 3 = 0
Using the Q Formula
-1 + sqrt [ 1 - 4* -3 * 5/6 ]
x = _______________________
2 (5/6)
x = -1 + sqrt [ 11]
_______________
5/3
-3 + 3sqrt (11)
x = _______________
5
acd / b = (-3) (11) (5) / 3 = -55
