Quadratic modulus equations against constants

For d 0, |q(x)|=d means q(x)=d or q(x)=-d. These are intersections of the inner quadratic with two horizontal levels. Solve each quadratic, reject nonreal values, remove duplicate roots, then apply any domain. At d=0, solve one quadratic only. For |x2-1|=1, the branches give x2=2 or x2=0. Thus x=- 2,0, 2: three distinct roots. Check: two sign branches do not guarantee four roots. A branch may miss the parabola or be tangent. For d<0 there are no roots.