How can I find the coordinates of crossing points between the line y=2x+1 and the curve y=x^2+x+1?
At the point of intersection, the ordinates of the two curves are equal .
So 2x+1 = x^2+x+1
x^2+x-1-2x-1 = 0
x^2-x = 0
x(x-1) = 0
x=0 or x= 1.
When x=0 , from y =2x+1 , we get y = 2*0 +1 =1
When x=1, y =2*1 +1 =3
(0 ,1) and (1, 3)
The coordinates of the intercepting point have to verify both, the equation of the line and the equation of the curve.
We'll put each factor as 0:
x=0 and x-1=0, x=1
Now, we'll substitute the x values in one of the 2 equation, and we'll do it in the linear equation,because it's more easy to calculate:
So the first intercepting point: A(0,1)
So the second intercepting point: B(1,3).