(4/(x-3))+(x/(3-x))=x

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Solution for (4/(x-3))+(x/(3-x))=x equation: 


D( x )

3-x = 0

x-3 = 0

3-x = 0

3-x = 0

3-x = 0 // - 3

-x = -3 // * -1

x = 3

x-3 = 0

x-3 = 0

x-3 = 0 // + 3

x = 3

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

4/(x-3)+x/(3-x) = x // - x

4/(x-3)+x/(3-x)-x = 0

(4*(3-x))/((x-3)*(3-x))+(x*(x-3))/((x-3)*(3-x))+(-x*(x-3)*(3-x))/((x-3)*(3-x)) = 0

4*(3-x)+x*(x-3)-x*(x-3)*(3-x) = 0

x^3+x^2-6*x^2-7*x+9*x+12 = 0

x^3-5*x^2+2*x+12 = 0

x^3-5*x^2+2*x+12 = 0

x^3-5*x^2+2*x+12 = 0

{ 1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 12, -12 }

1

x = 1

x^3-5*x^2+2*x+12 = 10

1

-1

x = -1

x^3-5*x^2+2*x+12 = 4

-1

2

x = 2

x^3-5*x^2+2*x+12 = 4

2

-2

x = -2

x^3-5*x^2+2*x+12 = -20

-2

3

x = 3

x^3-5*x^2+2*x+12 = 0

3

x-3

x^2-2*x-4

x^3-5*x^2+2*x+12

x-3

3*x^2-x^3

2*x-2*x^2+12

2*x^2-6*x

12-4*x

4*x-12

0

x^2-2*x-4 = 0

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

DELTA = 20

DELTA > 0

x = (20^(1/2)+2)/(1*2) or x = (2-20^(1/2))/(1*2)

x = (2*5^(1/2)+2)/2 or x = (2-2*5^(1/2))/2

x in { (2-2*5^(1/2))/2, (2*5^(1/2)+2)/2, 3} 

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

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

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

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

( x-((2*5^(1/2)+2)/2) )

x-((2*5^(1/2)+2)/2) = 0 // + (2*5^(1/2)+2)/2

x = (2*5^(1/2)+2)/2

( x-((2-2*5^(1/2))/2) )

x-((2-2*5^(1/2))/2) = 0 // + (2-2*5^(1/2))/2

x = (2-2*5^(1/2))/2

( x-3 )

x-3 = 0 // + 3

x = 3

x in { 3} 

x in { (2*5^(1/2)+2)/2, (2-2*5^(1/2))/2 }

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