Problem: Given z and 1/z are roots of the quadratic equation 3kx^2 -18tx + (2k+3) = 0 where t and k are constants. Find the value of k.

I attempted this by using the method of equating coefficients:

If z and 1/z are roots then it follows that (x-z) and (x-1/z) must be roots.
Multiplying these roots out gives:
x^2 - (z+1/z)x + 1 = 0

If I equate the coefficients of the two quadratic equations, I get:
3k = 1, (z+1/z) = 18, and 2k+3 = 1

Problem is this gives me two different values for k:
if 3k=1, then k = 1/3 .... but if 2k+3=1, then k= -1

However, if I use the following property of quadratic roots I get yet a different answer (the correct answer).
Property: product of the roots = c/a (for an equation of the form ax^2 + bx +c = 0)
This tells me that z * (1/z) = (2k+3)/3k
=> z/z = 1 = (2k+3)/3k
=> 2k+3 = 3k
=> k = 3

Going back to equating coefficients, I do get the answer k = 1 IF I FIRST DIVIDE BOTH SIDES OF 3kx^2 -18tx + (2k+3) = 0 by 3k to get the coefficient of the x squared term the same on both sides before I start equating coefficients.
Can anyone tell me why I have to do this first? Is there a rule that says you can only equate coefficients in quadratics if the coefficient of x squared is one or something similar to this? I've googled around a bit and I can't find any clear answer to this. Any help in understanding this would be greatly appreciated.