The gradient of a curve y= ax^2 +bx at the point (2,4) is -8, calculate the values of a and b
$\displaystyle \frac{dy}{dx} = 2ax + b$. Specifically, at (2,4), $\displaystyle \frac{dy}{dx} = 2a(2) + b = 4a + b = -8$.
However...we have one equation, two variables. Can't really find a unique solution (a,b). However, we do know that, for the original equation, when x = 2, y = 4.
$\displaystyle 4 = a(2^2) + b(2) = 4a + 2b$, so we have the system
$\displaystyle 4a + b = -8$
$\displaystyle 4a + 2b = 4$
Solve the system of equations.