Answer
The value of the expression $\frac{-b-\sqrt{{{b}^{2}}-4ac}}{2a}$ for $a=2,b=9,\text{ and }c=-5$ is $-5$.
Work Step by Step
Consider the expression, $\frac{-b-\sqrt{{{b}^{2}}-4ac}}{2a}$
Now, substitute the value $a=2,b=9,\text{ and }c=-5$ in the expression $\frac{-b-\sqrt{{{b}^{2}}-4ac}}{2a}$
Therefore,
$\begin{align}
& \frac{-b-\sqrt{{{b}^{2}}-4ac}}{2a}=\frac{-9-\sqrt{{{\left( 9 \right)}^{2}}-4\cdot 2\cdot \left( -5 \right)}}{2\cdot 2} \\
& =\frac{-9-\sqrt{81+40}}{4} \\
& =\frac{-9-\sqrt{121}}{4} \\
& =\frac{-9-11}{4}
\end{align}$
Further simplify,
$\begin{align}
& \frac{-b-\sqrt{{{b}^{2}}-4ac}}{2a}=\frac{-20}{4} \\
& =-5
\end{align}$
Therefore, the value of the expression $\frac{-b-\sqrt{{{b}^{2}}-4ac}}{2a}$ for $a=2,b=9,\text{ and }c=-5$ is $-5$.
