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

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At the point where the line y=3x+2 meets the curve g(x) = 2x^2 -3x-2, the values of x and y are the same.

We equate the two equations:

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

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

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

x1 = [-b + sqrt (b^2 - 4ac)]/2a

=> [ 3 + sqrt (9 +8)]/2

=> 3/2 + (sqrt 17) /2

x2 = 3/2 - (sqrt 17) /2

For x = 3/2 + (sqrt 17) /2, y = 13/ 2 + 3*(sqrt 17)/2

For x = 3/2 - (sqrt 17) /2, y = 13/ 2 - (3*sqrt 17)/2

**The points of intersection are (3/2 + (sqrt 17)/2 , 13/2 + 3*(sqrt 17)/2) and (3/2 - (sqrt 17)/2, 13/2 - 3*(sqrt 17)/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 )**