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Find the product of the y-coordinates of all the distinct solutions (x,y) for the two equations y = x^2 - 8 and y^2 = -6x + 40.

 Dec 21, 2020
 #1
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y = x^2 - 8  ⇒  y^2 = ( x^4  - 16x^2 + 64)      (1)         

 

y^2 = -6x  + 40     (2)

 

Sub (1)  into (2)   and we  have that

 

x^4 -16x^2  +  64 =  -6x  + 40      simplify

 

x^4  - 10x^2  + 24   = 0       factor as

 

(x^2  - 6)  ( x^2 - 4)   = 0

 

(x^2 - sqrt (6) ) ( x + sqrt (6)) ( x - 2) ( x + 2)   = 0

 

Setting each of these to 0 and solving for  x produces

 

x = sqrt (6)   x = -sqrt (6)    x = 2 and x = -2

 

And the associated   y  values will  be

 

y = [ ±sqrt (6) ]^2 - 8    =   6 - 8 =  -2   (twice)        and  y=  (±2)^2  - 8  =  4 - 8  =  -4  (twice)

 

So....the sum of these  =   -2 + -2 + -4 + -4  =  -12

 

 

cool cool cool

 Dec 22, 2020

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