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 functions =
x^2 + 4x + 6 = (1/2)x^2 + x + 4 simplify as
(1/2)x^2 + 3x + 2 = 0 multiply through by 2
x^2 + 6x + 4 = 0
x^2 + 6x = -4 complete the square on x
x^2 + 6x + 9 = -4 +9
(x + 3)^2 = 5 take both roots
x + 3 = sqrt 5 x + 3 = - sqrt (5)
x = sqrt (5) - 3 = c x = -sqrt (5) - 3 = a
c - a =
2 sqrt (5)
