MHB Find x for Geometric Progression: Solve with Step-by-Step Explanation

[Fernando Revilla]

I quote a question from Yahoo! Answers

Find the value of x such that the following sequence forms a geometric progression...?
x-1, 3x+4, 6x+8...so i am suppose to solve this by this rule: a,b,c then b^2=ac but I am just going around in circles because i have no idea how to get an answer, my textbook says the answer is -6, but i want to know the working out...any answers appreciated!
thanks in advance:)

I have given a link to the topic there so the OP can see my response.

 

[Fernando Revilla]

The sequence $x-1, 3x+4, 6x+8$ forms a a geometric progression if and only if:
$$\frac{3x+4}{x-1}=\frac{6x+8}{3x+4}\text{ and } x-1\neq 0\text{ and }3x+4\neq 0$$
Solving the equation
$$3x^2+22x+24=0\Leftrightarrow\ldots \Leftrightarrow x=-6\text{ or }x=-4/3$$
But $x=-4/3$ is not a valid solution (satisfies $3x+4=0$), so the solution is $x=-6$.