Question

When line y=2x+2 and parabola y= negative x squared - 4x-3 intersect at points A and B, find the length of the line segment AB
from kumon level I math page 192a

Answer 1

To find the points of intersection, we will set the two equations equal to each other:

2x+2 = -x^2 - 4x - 3

Simplifying, we get:

x^2 + 6x + 5 = 0

Factoring the quadratic equation, we get:

(x+1)(x+5) = 0

So, x = -1 or x = -5.

When x = -1, y = 2(-1) + 2 = 0
When x = -5, y = 2(-5) + 2 = -8

So the two points of intersection are (-1, 0) and (-5, -8).

To find the length of the line segment AB, we will use the distance formula:

AB = sqrt((x2 - x1)^2 + (y2 - y1)^2)

AB = sqrt((-5 - (-1))^2 + (-8 - 0)^2)
AB = sqrt((-4)^2 + (-8)^2)
AB = sqrt(16 + 64)
AB = sqrt(80)

Therefore, the length of the line segment AB is sqrt(80) or approximately 8.944.