### Instruction: Find the greatest value of $b$ such that $-b^2+7b-10 \ge 0$.### Response: Let's first look at the expression $-b^2+7b-10$.
Right. The first term is $-b^2$.
The coefficient of $b^2$ is $-1$, so the graph of the expression will be a parabola that opens downward.
That means the greatest value of $b$ will be the $x$-coordinate of the vertex of the parabola.
Right. So let's find the $x$-coordinate of the vertex.
The $x$-coordinate of the vertex is $-\frac{b}{2a}$.
In this case, $a=-1$ and $b=7$, so the $x$-coordinate of the vertex is $-\frac{7}{2(-1)}$.
Which is $\frac{7}{2}$.