On the Quaternionic Quadratic Equation \(\varvec{xax+bx+xc+d=0}\)

It is shown that the roots of the quaternion quadratic equation \(xax+bx+xc+d=0\) can be found by computing the eigenvalues of a companion matrix. This allows to completely discuss the number of solutions and to apply some Gersgorin type theorems. Another consequence is the possibility of constructing equations with a given number of solutions.