common pointsWhat is the common point of the line y=1+2x and the parabola y=x^2+x+1?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to find the common points of y=1+2x and the parabola y=x^2+x+1

substitute y = 2x + 1 in y=x^2+x+1

=> 2x + 1 = x^2 + x + 1

=> x^2 - x = 0

=> x(x - 1) = 0

=> x = 0 and x = 1

For x = 0 , y = 1 and for x = 1, y = 3

The common points are (0,1) and (1, 3)

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

The common point that lies on the line and parabola in the same time is the intercepting point of the line and parabola.

So, the y coordinate of the point verify the equation of the line and the equatin of the parabola, in the same time.

2x+1=x^2+x+1

We'll move all term to one side and we'll combine like terms:

x^2-x=0

We'll factorize by x:

x*(x-1)=0

We'll put each factor as zero:

x=0

x-1=0

We'll add 1 both sides:

x=1

Now, we'll substitute the value of x in the equation of the line, because it is much more easier to compute y.

y=2x+1

x=0

y=2*0+1, y=1

So the first pair of coordinates of crossing point: A(0,1)

x=1

y=2*1+1=3

So the second pair of coordinates of crossing point: B(1,3).

So, the common points are: A(0,1) and B(1,3).

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