  There are four division algebras over $\mathbb{R}$, namely real numbers,
complex numbers, quaternions, and octonions. Lack of commutativity and
associativity make it difficult to investigate algebraic and geometric
properties of octonions. It does not make sense to ask, for example, whether
the equation $x^2+1=0$ is solvable, without specifying the field in which we
want the solutions to be lie. The equation $x^2+1=0$ has no solutions in
$\mathbb{R}$, which is to say, there are no real numbers satisfying this
equation. On the other hand, there are complex numbers which do satisfy this
equation in the field $\mathbb{C}$ of all complex numbers. How about if we
extend the same idea to other two normed division algebras quaternions and
octonions. Liping Huang and Wasin So derive explicit formulas for computing the
roots of quaternionic quadratic equations. We extend their work to octonionic
case and solve monic left octonionic quadratic equation of the form
$x^2+bx+c=0$, where $a,b$ are octonions in general.[ We called this form of
quadratic equation as left octonion quadratic equation because we can consider
$x^2+xb+c=0$ as a different case due to non-commutativity of octonions].
Finally, we represent the left spectrum of $2\times2$ octonionic matrix as a
set of solutions to a corresponding octonionic quadratic equation, which is an
application of deriving explicit formulas for computing the roots of octonionic
quadratic equations.

General Mathematics