SOLUTION: 12(x^2+1/x^2)-56(x+1/x)+89=0

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Question 877681: 12(x^2+1/x^2)-56(x+1/x)+89=0

Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
12%28x%5E2%2B1%2Fx%5E2%29-56%28x%2B1%2Fx%29%2B89%22%22=%22%22%220%22
Let u%22%22=%22%22x%2B1%2Fx

Then

    u%5E2%22%22=%22%22%28x%2B1%2Fx%29%5E2%22%22=%22%22x%5E2%2B2%2B1%2Fx%5E2

    u%5E2%22%22=%22%22x%5E2%2B1%2Fx%5E2%2B2

    u%5E2-2%22%22=%22%22x%5E2%2B1%2Fx

So substitute u%5E2-2 for x%5E2%2B1%2Fx
and u for x%2B1%2Fx

12%28u%5E2-2%29-56u%2B89%22%22=%22%22%220%22

12u%5E2-24-56u%2B89%22%22=%22%22%220%22

12u%5E2-56u%2B65%22%22=%22%22%220%22

%286u-13%29%282u-5%29%22%22=%22%22%220%22

6u - 13 = 0;       2u - 5 = 0
     6u = 13           2u = 5
      u = 13%2F6         u =5%2F2

For u = 13%2F6, since u%22%22=%22%22x%2B1%2Fx

x%2B1%2Fx%22%22=%22%2213%2F6

Multiply through by 6x

6x%5E2%2B6%22%22=%22%2213x

6x%5E2-13x%2B6%22%22=%22%22%220%22

(3x-2)(2x-3) = 0

3x-2 = 0;     2x-3 = 0
  3x = 2;       2x = 3 
   x = 2%2F3;       x = 3%2F2


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For u = 5%2F2, since u%22%22=%22%22x%2B1%2Fx

x%2B1%2Fx%22%22=%22%225%2F2

Multiply through by 2x

2x%5E2%2B2%22%22=%22%225x

2x%5E2-5x%2B2%22%22=%22%22%220%22

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

2x-1 = 0;      x-2 = 0
  2x = 1;        x = 2 
   x = 1%2F2;       

So there are four solutions:

x = 2%2F3
x = 3%2F2
x = 1%2F2
x = 2

Edwin