Multiply each side of the equation by x, yielding bx^2 + cx = 0.  One root will be 0 from the extra x factor, and the other will be the root of the equation.

Apply the quadratic formula to produce: (-c +/- abs(c))/(2b).  For one sign of the root, the abs will be the same sign as -c, giving x=-c/b.  For the other root, the abs will be the opposite sign, giving x=0.

Since x=0 is the extra root, the root of bx+c=0 is -c/b.