Determine the real numbers a, c if the points (1;2) and (0;3) are on the graph of f(x)=ax^2+x+c.

If the graph of the function f(x) is passing through the
given points, then the coordinates of these points have to verify the expression of the
function.

The point A(1 , 2) is on the graph if and only
if:

f(1) = 2

We'll substitute
x by 1:

f(1) = a + 1 + c

a +
1 + c = 2

The point B(0 , 3) is on the graph if and only
if:

f(0) = 3

We'll substitute
x by 0:

f(0) =
c

c =
3

We'll substitute c = 3
in:

a + 1 + c = 2

a + 1 + 3 =
2

We'll subtract 4 both
sides:

a = 2 -
4

a =
-2

The expression of f(x) is:
f(x) = -2x^2 + x + 3