How can I prove that this matrix is diagonalizable?

Question

How can I prove that for all $a \in \mathbb{R}$ the following matrix is diagonalizable? I computed the characteristic polynomial, but I couldn't decompose it into linear factors. This is the polynomial: $t^2-3t+2-a^2$ What should I do next?

Here is the matrix: \begin{bmatrix} 2 & a \\ a & 1 \end{bmatrix}

Answer 1

Find the zeors of the polynomial. If there are two of them, you are done else you have to take a closer look at the eigenspace and check if it is two-dimensional.

Answer 2

The discriminant of the polynomial is $$ 3^2-4(2-a^2)=1+4a^2 $$ For all real values of $a$, the discriminant is positive. What does this tell you about the roots of the polynomial?

What is a sufficient criterion for diagonalizability? (In other words, you don't need to compute the eigenspaces, in this case.)