y=x-x^(1/2)

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Solution for y=x-x^(1/2) equation: 


D( x )

x < 0

x < 0

x in <0:+oo)

y = x-x^(1/2) // - x-x^(1/2)

x^(1/2)-x+y = 0

t_1 = x^(1/2)

1*t_1^1-1*t_1^2+y = 0

t_1-t_1^2+y = 0

DELTA = 1^2-(-1*4*y)

DELTA = 4*y+1

4*y+1 = 0

4*y+1 = 0 // - 1

4*y = -1 // : 4

y = -1/4

DELTA = 0  <=>  t_2 = -1/4

t_1 = -1/(-1*2) i y = -1/4

t_1 = 1/2 i y = -1/4

( t_1 = ((4*y+1)^(1/2)-1)/(-1*2) or t_1 = (-(4*y+1)^(1/2)-1)/(-1*2) ) i y > -1/4

( t_1 = ((4*y+1)^(1/2)-1)/(-2) or t_1 = ((4*y+1)^(1/2)+1)/2 ) i y > -1/4

y-(-1/4) > 0

y+1/4 > 0

y+1/4 > 0 // - 1/4

y > -1/4

t_1 = 1/2

x^(1/2)-1/2 = 0

1*x^(1/2) = 1/2 // : 1

x^(1/2) = 1/2

x^(1/2) = 1/2 // ^ 2

x = 1/4

t_1 = ((4*y+1)^(1/2)-1)/(-2)

t_3 = -(((4*y+1)^(1/2)-1)/(-2))

t_3+x^(1/2) = 0

1*x^(1/2) = -t_3 // : 1

x^(1/2) = -t_3

x^(1/2) = -t_3 // ^ 2

x = t_3^2

t_1 = ((4*y+1)^(1/2)+1)/2

t_4 = -(((4*y+1)^(1/2)+1)/2)

t_4+x^(1/2) = 0

1*x^(1/2) = -t_4 // : 1

x^(1/2) = -t_4

x^(1/2) = -t_4 // ^ 2

x = t_4^2

x in { 1/4, (-(((4*y+1)^(1/2)-1)/(-2)))^2, (-(((4*y+1)^(1/2)+1)/2))^2 }

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