# How can I find the coordinates of crossing points between the line y=2x+1 and the curve y=x^2+x+1?

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The coordinates of the intercepting point have to verify both, the equation of the line and the equation of the curve.

x^2-x=0

We'll factorize:

x*(x-1)=0

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:

y=2x+1

x=0

y=2*0+1, y=1

**So the first intercepting point: A(0,1) **

x=1

y=2*1+1=3

**So the second intercepting point: B(1,3).**

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

So

(0 ,1) and (1, 3)