(5/(2x+3))+(4/(2x-3))=((14x+3)/(4x^2-9))

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Solution for (5/(2x+3))+(4/(2x-3))=((14x+3)/(4x^2-9)) equation: 


D( x )

2*x-3 = 0

4*x^2-9 = 0

2*x+3 = 0

2*x-3 = 0

2*x-3 = 0

2*x-3 = 0 // + 3

2*x = 3 // : 2

x = 3/2

4*x^2-9 = 0

4*x^2-9 = 0

4*x^2 = 9 // : 4

x^2 = 9/4

x^2 = 9/4 // ^ 1/2

abs(x) = 3/2

x = 3/2 or x = -3/2

2*x+3 = 0

2*x+3 = 0

2*x+3 = 0 // - 3

2*x = -3 // : 2

x = -3/2

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

5/(2*x+3)+4/(2*x-3) = (14*x+3)/(4*x^2-9) // - (14*x+3)/(4*x^2-9)

5/(2*x+3)+4/(2*x-3)-((14*x+3)/(4*x^2-9)) = 0

5/(2*x+3)+4/(2*x-3)+(-1*(14*x+3))/(4*x^2-9) = 0

(5*(2*x-3)*(4*x^2-9))/((2*x+3)*(2*x-3)*(4*x^2-9))+(4*(2*x+3)*(4*x^2-9))/((2*x+3)*(2*x-3)*(4*x^2-9))+(-1*(14*x+3)*(2*x+3)*(2*x-3))/((2*x+3)*(2*x-3)*(4*x^2-9)) = 0

5*(2*x-3)*(4*x^2-9)+4*(2*x+3)*(4*x^2-9)-1*(14*x+3)*(2*x+3)*(2*x-3) = 0

72*x^3-56*x^3-12*x^2-12*x^2-162*x+126*x+27+27 = 0

16*x^3-24*x^2-36*x+54 = 0

16*x^3-24*x^2-36*x+54 = 0

2*(8*x^3-12*x^2-18*x+27) = 0

8*x^3-12*x^2-18*x+27 = 0

{ 1, -1, 3, -3, 9, -9, 27, -27 }

1

x = 1

8*x^3-12*x^2-18*x+27 = 5

1

-1

x = -1

8*x^3-12*x^2-18*x+27 = 25

-1

3

x = 3

8*x^3-12*x^2-18*x+27 = 81

3

-3

x = -3

8*x^3-12*x^2-18*x+27 = -243

-3

9

x = 9

8*x^3-12*x^2-18*x+27 = 4725

9

-9

x = -9

8*x^3-12*x^2-18*x+27 = -6615

-9

27

x = 27

8*x^3-12*x^2-18*x+27 = 148257

27

-27

x = -27

8*x^3-12*x^2-18*x+27 = -165699

-27

{ 1/2, -1/2, 1/4, -1/4, 1/8, -1/8, -1/2, 1/2, -1/4, 1/4, -1/8, 1/8, 3/2, -3/2, 3/4, -3/4, 3/8, -3/8, -3/2, 3/2, -3/4, 3/4, -3/8, 3/8, 9/2, -9/2, 9/4, -9/4, 9/8, -9/8, -9/2, 9/2, -9/4, 9/4, -9/8, 9/8, 27/2, -27/2, 27/4, -27/4, 27/8, -27/8, -27/2, 27/2, -27/4, 27/4, -27/8, 27/8 }

1/2

x

1/2

8*x^3-12*x^2-18*x+27 = 16

1/2

-1/2

x

-1/2

8*x^3-12*x^2-18*x+27 = 32

-1/2

1/4

x

1/4

8*x^3-12*x^2-18*x+27 = 21.875

1/4

-1/4

x

-1/4

8*x^3-12*x^2-18*x+27 = 30.625

-1/4

1/8

x

1/8

8*x^3-12*x^2-18*x+27 = 24.578125

1/8

-1/8

x

-1/8

8*x^3-12*x^2-18*x+27 = 29.046875

-1/8

-1/2

x

-1/2

8*x^3-12*x^2-18*x+27 = 32

-1/2

1/2

x

1/2

8*x^3-12*x^2-18*x+27 = 16

1/2

-1/4

x

-1/4

8*x^3-12*x^2-18*x+27 = 30.625

-1/4

1/4

x

1/4

8*x^3-12*x^2-18*x+27 = 21.875

1/4

-1/8

x

-1/8

8*x^3-12*x^2-18*x+27 = 29.046875

-1/8

1/8

x

1/8

8*x^3-12*x^2-18*x+27 = 24.578125

1/8

3/2

x

3/2

8*x^3-12*x^2-18*x+27 = 0

3/2

x-3/2

8*x^2-18

8*x^3-12*x^2-18*x+27

x-3/2

12*x^2-8*x^3

27-18*x

18*x-27

0

8*x^2-18 = 0

DELTA = 0^2-(-18*4*8)

DELTA = 576

DELTA > 0

x = (576^(1/2)+0)/(2*8) or x = (0-576^(1/2))/(2*8)

x = 3/2 or x = -3/2

x in { -3/2, 3/2, 3/2} 

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

(2*(x+3/2)*(x-3/2)^2)/((2*x+3)*(2*x-3)*(4*x^2-9)) = 0

(2*(x+3/2)*(x-3/2)^2)/((2*x+3)*(2*x-3)*(4*x^2-9)) = 0 // * (2*x+3)*(2*x-3)*(4*x^2-9)

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

( x+3/2 )

x+3/2 = 0 // - 3/2

x = -3/2

( x-3/2 )

x-3/2 = 0 // + 3/2

x = 3/2

x in { -3/2} 

x in { 3/2} 

x belongs to the empty set

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