Question

The curve y=ax^2+bx+c passes through the points (1, 0), (2, 0) and its gradient at the point (2, 0) is 2. Find the numerical value of the area included between the curve and the x-axis.

Answer 1

Given that the curve passes through (1,0) and (2,0), you know that
0 = a(2)² + b(2) + c
0 = a(1)² + b(1) + c

Or, equivalently,
4a + 2b + c = 0
a + b + c = 0

You're also given the slope of the curve at (2,0) to be 2, which means y'(2) = 2.
If y = ax² + bx + c, then y' = 2ax + b, and so you have the equation
2 = 2a(2) + b, or
2 = 4a + b

So you have three equations with three unknowns:
4a + 2b + c = 0
a + b + c = 0
4a + b = 2

First, solve for the unknowns, then find the area of the desired region.

Answer 2

Thank you!