Real matrix/2x2/Trigonalizability via characteristic polynomial/Example

We consider a real ${}2\times 2$-matrix ${}M={\begin{pmatrix}a&b\\c&d\end{pmatrix}}$. The characteristic polynomial is

${}{\begin{aligned}\chi _{M}&=\det {\left(xE_{2}-M\right)}\\&=\det {\begin{pmatrix}x-a&-b\\-c&x-d\end{pmatrix}}\\&=(x-a)(x-d)-bc\\&=x^{2}-(a+d)x+ad-bc\\&={\left(x-{\frac {a+d}{2}}\right)}^{2}-{\left({\frac {a+d}{2}}\right)}^{2}+ad-bc\\&={\left(x-{\frac {a+d}{2}}\right)}^{2}-{\left({\frac {a-d}{2}}\right)}^{2}-bc.\end{aligned}}$

This polynomial splits into (real) linear factors if and only if ${}{\left({\frac {a-d}{2}}\right)}^{2}+bc\geq 0$. The matrix is trigonalizable exactly in this case, due to fact.