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

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

Fun problem...!

Here is the key:

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.

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

or

Solve each of those equations....

or

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.

or

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.

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