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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?

 Jul 6, 2022
 #1
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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)

 

cool cool cool

 Jul 6, 2022

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