three numbers with the product of -24 and a sum of 1

Answer 1

xyz = -24

x+y+z = 1

x + (-24/(xz)) + z = 1

x^2z - 24 + xz^2 = xz

x^2z + xz^2 - xz = 24

x^2z + x( z^2 - z) = 24

z * x^2 + (z^2 - z)*x - 24 = 0

this is quadratic equation in terms of x... A = z, B = (z^2 -z) , C = -24

x =[ (z - z^2) +or- sqrt( (z^2 - z)^2 + 4(z)(24))]/ (2z)

x = [ (z - z^2) +or- sqrt( (z^2 - z)^2 + 96z) ]/ (2z)

z is the free variable

once chosen , determines the value of x, which then will get you the value of y= -24/(xz)

For example:

if z=1

then x = +or- sqrt(96)/2 = +or- 4*sqrt(6)/2 = +or- 2* sqrt(6)

So then y = -24/(+or-2 * sqrt(6)) = -or+ 24*sqrt(6) / 12 = -or+ 2*sqrt(6)

THeir sum is in fact 1, and their product is 24