Question

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 3x-1. Solve for (x) to determine the lengths of the two internal segments of the secants.

Answer 1

I assume you meant:
"...and the <i>second</i> internal segment length is 3x-1..."

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+3x-1=14+3x

From equation (1),
12(13+3x)=15(14+3x)
which gives
x=-54/9=-6, or
3x+1=-17, 3x-1=-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.