= Vertex form of a quadratic
{title2=$h=-b/(2a),\quad k=c-b^2/(4a)$}
= Completed-square form
{synonym}
Write the <quadratic polynomial> as
$$
p(x)=a(x-h)^2+k,\qquad h=-\frac b{2a},\qquad k=c-\frac{b^2}{4a}.
$$
Its <function graph> is a <parabola> with <vertex of a parabola> $(h,k)$. Equal offsets from $h$ give equal values, since $p(h+t)=p(h-t)=at^2+k$. If $a>0$, the <minimum> is $k$; if $a<0$, the <maximum> is $k$. In both cases it is attained only at $x=h$.
For the nonmonic <quadratic polynomial>
$$
-2x^2+8x-5=-2(x^2-4x)-5
=\boxed{-2(x-2)^2+3},
$$
the negative <leading coefficient> reverses the <inequality>: $-2(x-2)^2\leq0$. Its <maximum> is $3$ at $x=2$. Its <coefficients> are different from those of the positive example, but the same form exposes the shape of both <function graphs>.
\Image[/completing-square-vertices-and-real-roots.png]
{title=Completed squares reveal a quadratic's vertex, extremum and real roots}
{description=Original <function graphs> of the two stipulated <quadratic polynomials>. On the left, $(x+3)^2+4$ stays above zero. On the right, $-2(x-2)^2+3$ opens downward and has <roots of a polynomial> $2\pm\sqrt{3/2}$; the green labels round these values to three decimal places. The dashed lines pass through the <vertices of parabolas>.}
Completing the square also simplifies an <integral> containing the <exponential function>:
$$
\int_{-\infty}^{\infty}e^{-(x^2+6x+13)}\,dx
=e^{-4}\int_{-\infty}^{\infty}e^{-(x+3)^2}\,dx
=e^{-4}\sqrt\pi.
$$
The substitution $u=x+3$ removes the linear term and leaves the standard <Gaussian integral>. The same shift that locates the <minimum> makes this <integral> recognizable.
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