When line y=2x+3 and parabola y=x squared intersect at points A and B, find the length of the line segment AB
To find the points of intersection between the line y = 2x + 3 and the parabola y = x^2, we set the two equations equal to each other:
2x + 3 = x^2
Rearranging the equation, we get:
x^2 - 2x - 3 = 0
Solving the quadratic equation using the quadratic formula, we get:
x = (2 ± √(2^2 - 4(1)(-3))) / 2
x = (2 ± √(4 + 12)) / 2
x = (2 ± √16) / 2
x = (2 ± 4) / 2
Therefore, x = -1 or x = 3
Substitute x = -1 and x = 3 back into the equation y = 2x + 3 to find the corresponding y values:
For x = -1, y = 2(-1) + 3 = 1
For x = 3, y = 2(3) + 3 = 9
So the two points of intersection are (-1, 1) and (3, 9).
Using the distance formula, we find the length of the line segment AB:
AB = √((3 - (-1))^2 + (9 - 1)^2)
AB = √(16 + 64)
AB = √80
AB = 4√5
Therefore, the length of the line segment AB is 4√5 units.