Answer
$(-5,-2)$
Work Step by Step
To find the vertex, the given function, $
f(x)=x^2+10x+23
,$ should be converted in the form $f(x)=a(x-h)^2+k$.
Grouping the variables together, the given function is equivalent to
\begin{align*}
f(x)&=(x^2+10x)+23
.\end{align*}
Completing the square of the expression with variables by adding $\left(\dfrac{b}{2}\right)^2,$ the expression above is equivalent to
\begin{align*}
f(x)&=\left(x^2+10x+\left(\dfrac{10}{2}\right)^2\right)+\left[23-\left(\dfrac{10}{2}\right)^2\right]
\\\\
f(x)&=\left(x^2+10x+25\right)+\left[23-25\right]
\\\\
f(x)&=\left(x+5\right)^2-2
.\end{align*}Note that $\left(\dfrac{b}{2}\right)^2$ should be subtracted as well to negate the addition of that value when completing the square of the expression with variables.
In the form $f(x)=a(x-h)^2+k$, the equation above is equivalent to
\begin{align*}
f(x)&=\left(x-(-5)\right)^2-2
.\end{align*}
Since the vertex of the quadratic function $f(x)=a(x-h)^2+k$ is given by $(h,k)$, then the vertex of the quadratic function above is $
(-5,-2)
$.
