when line y=x+2 and parabola y=x squared intersect at points A and B, find the length of the line segment AB
To find the intersection points of the line y=x+2 and the parabola y=x^2, we need to solve the system of equations.
Setting y=x+2 equal to y=x^2, we get:
x + 2 = x^2
Rearranging the equation, we get:
x^2 - x - 2 = 0
This is a quadratic equation that can be factored as:
(x - 2)(x + 1) = 0
So, the solutions are x = 2 and x = -1.
Substitute these values back into the equation y=x+2:
When x = 2, y = 2+2 = 4 (point A)
When x = -1, y = -1+2 = 1 (point B)
The length of the line segment AB can be found using the distance formula:
AB = sqrt((x2-x1)^2 + (y2-y1)^2)
AB = sqrt((2-(-1))^2 + (4-1)^2)
AB = sqrt(3^2 + 3^2)
AB = sqrt(18)
AB = 3√2
Therefore, the length of the line segment AB is 3√2.