Geometry
posted by Carolina .
Two secants are drawn to a circle from an external point. The external secant length on the first secant is 12 and the internal segment length is 3x +1. The external secant length on the second secant is 15 and the two internal segment length is 3x1. Solve for (x) to determine the lengths of the two internal segments of the secants.

I assume you meant:
"...and the second internal segment length is 3x1..."
First, recall that a secant from an external point P cutting a given circle at A and B has the following properties:
PA*PB = PA'*PB' = PT^2....(1)
where A'B' are two other points of intersection of a second secant, and T is the point of tangency.
For the given case,
PA=12, PB=12+3x+1=13+3x
PA'=15, PB'=15+3x1=14+3x
From equation (1),
12(13+3x)=15(14+3x)
which gives
x=54/9=6, or
3x+1=17, 3x1=19 (impossible).
Unless I made a mistake somewhere, I suggest you recheck the question and proceed to solve for x in the same way above.