Suppose that there is only one value of x for which the distance from (5,6) to (3x-7,ax+2) is 4. If a doesn't equal 0, what is a?

You get a quadratic, and you need the discriminant to be 0.  This gives you the equation

72^2 - 144*8a - 4*3*(a + 9)*144 = 0.

 

Solution is a = -18/5