Odd Middle Terms

When the middle number is odd, half of it is a fraction, and nothing else changes. Take ${x}^{2}+5x+2$. Half of 5 is 5/2, and its square is 25/4. Add and subtract it: ${x}^{2}+5x+\frac{25}{4}-\frac{25}{4}+2$.

The first three terms are ${\left(x+\frac{5}{2}\right)}^{2}$. For the rest, write 2 as 8/4: $\frac{-25}{4}+\frac{8}{4}$ is -17/4. So the expression is ${\left(x+\frac{5}{2}\right)}^{2}-\frac{17}{4}$, and the vertex is (-5/2, -17/4).

Keep the fractions exact. The vertex form must equal the original for every x, so rounding 25/4 to 6.3 would quietly make it a different expression.