Prove that the curves x^2-3x+1 and 2x^2+x+4 are intercepting.

To prove that 2 curves are intercepting, we'll have to
solve the system formed by the equations of the curves. If the curves are intercepting,
then the system will have solutions.

We'll note y =
x^2-3x+1 (1)

 and y = 2x^2+x+4
(2)

We'll put (1) =
(2)

x^2-3x+1 = 2x^2+x+4

We'll
subtract boths sides x^2-3x+1 and we'll use symmetric
property:

x^2 + 4x + 3 =
0

We'll apply quadratic
formula:

x1 = [-4 + sqrt(16 -
12)]/2

x1 = (-4+2)/2

x1 =
-1

x2 = (-4-2)/2

x2 =
-3

For x1 = -1, y1 = 1+3+1 =
5

For x2 = -3, y = 9+9+1 =
19

The intercepting points of the given
curves are: (-1 ; 5) and (-3 ; 19).