Question

The curve with equation y = ax^2 + bx + c passes through the points P(2,6) abd Q(3,16), and has a gradient of 7 at the point P. Find the values of the constants a,b and c.

Thanks!

Answer 1

These conditions must be satisfied:
6 = a*2^2 + b*2 + c = 4a + 2b + c
16 = 9a + 3b + c
dy/dx @ P = 2a*2 + b = 4a + b = 7

Solve those three equations in 3 unknowns. It does not require integral calculus.
4a + 2b + c = 6
4a + b = 7
Combine these two to get b + c = -1

36a + 12 b + 4c = 64
36a + 18 b + 9c = 54
Combine the last 2 to get
6b + 5c = -10
Now you have two equations in two unknowns. Use substitution to eliminate one of the variables.
6b + 5(-1 -b) = -10
b = -5
4a = 7-b = 12 ; a = 3
5c = -10 -6b = -10 + 30 = 20
c = 4
y = 3x^2 -5x + 4