(18x^3+18x^2-2x+1)/(6x^2+4x)

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Solution for (18x^3+18x^2-2x+1)/(6x^2+4x) equation: 


D( x )

6*x^2+4*x = 0

6*x^2+4*x = 0

6*x^2+4*x = 0

6*x^2+4*x = 0

DELTA = 4^2-(0*4*6)

DELTA = 16

DELTA > 0

x = (16^(1/2)-4)/(2*6) or x = (-16^(1/2)-4)/(2*6)

x = 0 or x = -2/3

x in (-oo:-2/3) U (-2/3:0) U (0:+oo)

t_2 = 0

(18*x^3+18*x^2-2*x+1)/(6*x^2+4*x)+t_2 = 0

6*x^2+4*x = 0

6*x^2+4*x = 0

2*x*(3*x+2) = 0

3*x+2 = 0 // - 2

3*x = -2 // : 3

x = -2/3

2*x*(x+2/3) = 0

(18*x^3+18*x^2-2*x+1)/(2*x*(x+2/3))+t_2 = 0

(18*x^3+18*x^2-2*x+1)/(2*x*(x+2/3))+(2*t_2*x*(x+2/3))/(2*x*(x+2/3)) = 0

2*t_2*x*(x+2/3)+18*x^3+18*x^2-2*x+1 = 0

2*t_2*x^2+4/3*t_2*x+18*x^3+18*x^2-2*x+1 = 0

2*t_2*x^2+4/3*t_2*x+18*x^3+18*x^2-2*x+1 = 0

2*t_2*x^2+4/3*t_2*x+18*x^3+18*x^2-2*x+1

2*x*(9*x^2-1)+2*t_2*x^2+4/3*t_2*x+18*x^2+1

2*x*(2/3*t_2+9*x)+2*x*(9*x^2-1)+2*t_2*x^2+1

2*x*(2/3*t_2+9*x^2+9*x-1)+2*t_2*x^2+1

2*(2/3*t_2+9*x^2+9*x-1)

(2*(2/3*t_2+9*x^2+9*x-1))/(2*x*(x+2/3)) = 0

(2*(2/3*t_2+9*x^2+9*x-1))/(2*x*(x+2/3)) = 0 // * 2*x*(x+2/3)

2*(2/3*t_2+9*x^2+9*x-1) = 0

2/3*t_2+9*x^2+9*x-1 = 0

DELTA = 9^2-(4*9*(2/3*t_2-1))

DELTA = 81-36*(2/3*t_2-1)

81-36*(2/3*t_2-1) = 0

81-36*(2/3*t_2-1) = 0

117-24*t_2 = 0

117-24*t_2 = 0

117-24*t_2 = 0 // - 117

-24*t_2 = -117 // : -24

t_2 = -117/(-24)

t_2 = 39/8

DELTA = 0  <=>  t_4 = 39/8

x = -9/(2*9) i t_2 = 39/8

x = -1/2 i t_2 = 39/8

( x = ((81-36*(2/3*t_2-1))^(1/2)-9)/(2*9) or x = (-(81-36*(2/3*t_2-1))^(1/2)-9)/(2*9) ) i t_2 > 39/8

( x = ((81-36*(2/3*t_2-1))^(1/2)-9)/18 or x = (-((81-36*(2/3*t_2-1))^(1/2)+9))/18 ) i t_2 > 39/8

t_2-39/8 > 0

t_2-39/8 > 0 // + 39/8

t_2 > 39/8

x in { -1/2, ((81-36*(0*2/3-1))^(1/2)-9)/18, (-((81-36*(0*2/3-1))^(1/2)+9))/18 }

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