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

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The points (1 , 2) and (0,3) lie on the graph of the function f(x) = ax^2 + x + c.

This gives two equations:

2 = a*1 + 1 + c ...(1)

3 = 0 + 0 + c ...(2)

From (2) we get c = 3

substitute in (1)

2 = a + 1 + 3

=> a = -2

**The values of a and c are -2 and 3 respectively.**

Since the graph of the function f(x) is passing through the given points, then the coordinates of these points must 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**