Relation to the quadratic formula

Normalize the characteristic polynomial to $x^2-sx+p$ with root sum $s$ and root product $p$. Vieta says the roots average to $s/2$ and differ by a square root term .

The mean-product formula is not a different theorem: it is the quadratic formula rewritten so each symbol is readable from a $2\times 2$ matrix. You trade memorizing a generic quadratic template for memorizing trace, determinant, and one square root.

When the leading coefficient is already $1$, the product of roots is the constant term with no extra scaling. That is why determinant alone supplies $p$ in the eigenvalue story.

If you already know the roots of a monic quadratic, read off $s$ as their sum and $p$ as their product; the mean-product formula is the same information packaged for $2\times2$ matrices.

Check your understanding. The tasks below rest on these ideas: Correct: by Vieta's formulas, the root sum is the negative of the linear coefficient, here $s$. Not quite: $p$ is the product, and the other options are wrong. Correct: the product of the roots equals the constant term $p$. Not quite: $s$ is the sum, and $s^2$ or $s/2$ are unrelated. Correct: the roots are symmetric about $s/2$. Not quite: the mean is half the sum, not the product or a root of it. Correct: the trick simply reads the same quadratic's roots from its trace and determinant. Not quite: the determinant is not always zero, eigenvalues are not the diagonal in general, and trace and determinant differ.