If a, b and c are integers, then the discriminant of ${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}$ is of the form (where k is an integer)

b is even i.e b = 2m.
when ${\mathrm{b}}^2-4\mathrm{ac}=4\left({\mathrm{m}}^2-\mathrm{ac}\right)=4\mathrm{k}$
b is odd i.e , b = 2m -1 .
Then ${\mathrm{b}}^2-4\mathrm{ac}=(2\mathrm{m}-1{)}^2-4\mathrm{ac}$
$\begin{array}{l}=4{\mathrm{m}}^2+4\mathrm{m}+1-4\mathrm{ac}\\ =4\left({\mathrm{m}}^2+\mathrm{m}-\mathrm{ac}\right)+1=4\mathrm{k}+1\end{array}$