Answer
$x=-5$
Work Step by Step
Multiplying both sides by $2,$ the given expression is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{1}{2}x^2+5x+\dfrac{25}{2}=0
\\\\
2\left( \dfrac{1}{2}x^2+5x+\dfrac{25}{2} \right)=2(0)
\\\\
x^2+10x+25=0
.\end{array}
Using the factoring of trinomials in the form $x^2+bx+c,$ the $\text{
equation
}$
\begin{array}{l}\require{cancel}
x^2+10x+25=0
\end{array} has $c=
25
$ and $b=
10
.$
The two numbers with a product of $c$ and a sum of $b$ are $\left\{
5,5
\right\}.$ Using these two numbers, the $\text{
equation
}$ above is equivalent to
\begin{array}{l}\require{cancel}
(x+5)(x+5)=0
.\end{array}
Equating each factor to zero (Zero Product Property), and then isolating the variable, the solution is $
x=-5
.$
