At what points does the line y=3x+2 meet the curve g(x) = 2x^2 -3x-2 ?

We have the line y= 3x+2 and the curve g(x) = 2x^2
-3x-2

We need to find the points of intersection between
g(x) and the line y.

Then, we know that the points of
intersection must verify the equations of both
lines.

==> g(x) =
y

==> 2x^2 -3x -2 =
3x+2

Now we will combine all terms on the left
side.

==> 2x^2 -6x -4 =
0

We will divide by
2.

==> x^2 -3x -2 =
0

Now we will use the
formula.

==> x1= (3+ sqrt(9+8)/ 2= ( 3+
sqrt17)/2==> y1= 3x+2 = (9+3sqrt17)/2 + 2 =
(13+3sqrt17)/2

==> x2= (3-sqrt17)/2==> y2=
3x2+2 = (9-3sqrt17)/2 +2 = (13-3sqrt17)/2

Then the points
of intersection are:

( (3+sqrt17)/2 ,
(13+3sqrt17)/2 ) and ( (3-sqrt17)/2 , (13-3sqrt17)/2
)