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.
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