Partial Derivatives & the Gradient · Challenge problem

Let $r(b,c)$ be the larger root of $x^2 + bx + c = 0$, so $r(-5, 6) = 3$. Without using the quadratic formula, find $r_b$ and $r_c$ at $(b,c) = (-5, 6)$ and use the linear approximation of $r$ to estimate the larger root of $x^2 - 5.1x + 6.2 = 0$.