The parabolas defined by the equations y = x^2 + 4x + 6 and y = 1/2*x^2 + x + 4 intersect at (a,b) and (c,d) where c >= a. What is c - a?
+128408 Set the y's equal and we get that
x^2 + 4x + 6 = (1/2)x^2 + x + 4 rearrange as
(1 -1/2)x^2 + (4 -1)x + 6 - 4 = 0 simplify
(1/2)x^2 + 3x + 2 = 0 multiply through by 2
x^2 + 6x + 4 = 0
We only care about the x values = c , a
c = -6 +√[ 6^2 - 4(1)(4) ] -6 + √ 20
_________________ = _________ = -3 + √5
2 2
a is the conjugate to c = -6 - √20
___________ = -3 - √ 5
2
So
c - a = (-3 + √5 ) - ( -3 -√5 ) = 2√5
