SOLUTION: The curve y = ax^2 + bx has gradient 8 when x = 2 and has gradient -10 when x = -1. Find the value of a and the value of b.

The problem says that the derivative    is   8  when x= 2,    (1)

                  and the derivative    is -10  when x= -1.   (2)


So, we calculate the derivative   = 2ax + b,

and form two equations aka (1) and (2) for points x= 2  and  x= -1

    
    2a*2    + b =   8     (3)

    2a*(-1) + b = -10     (4)


Equivalently, this system of equations is

     4a + b =   8         (3')

    -2a + b = -10         (4')


To find "a" from these equations, we subtract equation (4') from equation (3'), making Elimination.

We get then

     4a - (-2a) = 8 - (-10)

        6a      =   18

         a      =   18/6 = 3.


Then from equation (3')

     b = 8 - 4a = 8 - 4*3 = 8 - 12 = -4.



ANSWER.  a = 3;  b = -4.