Let A be a 2x2 matrix and let

char poly(A) = z^2 - Tz + D = 0.

T=trace(A) D=det(A)

Show that all eigenvalues of A have nonzero real part if T*D /= 0

Show that all eigenvalues of A have real part < 0 if D > 0 and T < 0

Based on the characteristic polynomial of A, I know that

T= a1 + a2, where a1 and a2 are eigenvalues of A

D=a1*a2

However, I do not know how to proceed from here. Any help would be greatly appreciated!!