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

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Question 1187475: 12(x^2+1/x^2)-56(x+1/x)+89=0
Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


Fun problem...!

Here is the key:

x%5E2%2B1%2Fx%5E2+=+%28x%5E2%2B2%2B1%2Fx%5E2%29-2+=+%28x%2B1%2Fx%29%5E2-2

Use that to rewrite the first expression in parentheses and you end up with a quadratic equation with "x+1/x" as the "variable".

Solve that equation for x+1/x and you end up solving two quadratic equations that each have two real solutions, giving you four solutions to the given equation.

12%28x%5E2%2B1%2Fx%5E2%29-56%28x%2B1%2Fx%29%2B89=0
12%28x%5E2%2B2%2B1%2Fx%5E2%29-12%282%29-56%28x%2B1%2Fx%29%2B89=0
12%28x%2B1%2Fx%29%5E2-24-56%28x%2B1%2Fx%29%2B89=0
12%28x%2B1%2Fx%29%5E2-56%28x%2B1%2Fx%29%2B65=0

That's a quadratic equation in the "variable" x+1/x -- solve it by factoring.

%282%28x%2B1%2Fx%29-5%29%286%28x%2B1%2Fx%29-13%29=0

x%2B1%2Fx=5%2F2 or x%2B1%2Fx=13%2F6

Solve each of those equations....

x%2B1%2Fx=5%2F2
2x%2B2%2Fx=5
2x-5%2B2%2Fx=0
2x%5E2-5x%2B2=0
%282x-1%29%28x-2%29=0
x=1%2F2 or x=2

There are two solutions; note that they are reciprocals of each other. That makes sense, since the equation can be written as a quadratic with "variable" (x+1/x) -- if x=2 then 1/x=1/2; and if x=1/2 then 1/x=2.

x%2B1%2Fx=13%2F6
6x%2B6%2Fx=13
6x-13%2B6%2Fx=0
6x%5E2-13x%2B6=0
%282x-3%29%283x-2%29=0
x=3%2F2 or x=2%2F3

There are two more solutions which are also reciprocals of each other.

ANSWER: There are 4 solutions: x=2 or x=1/2; and x=3/2 and x=2/3

Here is a graph of the given equation, showing the four zeros at 1/2, 2/3, 3/2, and 2.

graph%28600%2C200%2C-1%2C4%2C-1%2C1%2C12%28x%5E2%2B1%2Fx%5E2%29-56%28x%2B1%2Fx%29%2B89%29