SOLUTION: sorry I already ask this question but i cannot find it again (x+ 1/x)^2 -2(x+ 1/x)+1=0 this part of equation in quadratic form I already got an answer x^4-2x^3+3x^2-2x+1=0 But

Question 210323This question is from textbook COLLEGE ALGEBRA
: sorry I already ask this question but i cannot find it again
(x+ 1/x)^2 -2(x+ 1/x)+1=0 this part of equation in quadratic form
I already got an answer x^4-2x^3+3x^2-2x+1=0 But i got stuck here i dont know what to do next
thank you for your help This question is from textbook COLLEGE ALGEBRA

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Start with the given equation.

Take note that the term is being repeated here. So one way to simplify this equation (for now) is to do a simple substitution

So let . So this means that

Replace each instance of with 'z'

Now we have a much simpler equation to solve for.

Notice that the quadratic is in the form of where , , and

Let's use the quadratic formula to solve for "z":

Start with the quadratic formula

Plug in , , and

Negate to get .

Square to get .

Multiply to get

Subtract from to get

Multiply and to get .

Take the square root of to get .

or Break up the expression.

or Combine like terms.

or Simplify.

So the only solution in terms of 'z' is

However, we want to solve for 'x' (not 'z'). But do recall that we let

Start with the given equation.

Plug in the solution (in terms of 'z')

Multiply EVERY term by the LCD 'x' to clear out the fraction.

Subtract 'x' from both sides.

Notice that the quadratic is in the form of where , , and

Let's use the quadratic formula to solve for "x":

Start with the quadratic formula

Plug in , , and

Negate to get .

Square to get .

Multiply to get

Subtract from to get

Multiply and to get .

Simplify the square root

or Break up the expression.

============================================================================
Answer:

So the solutions (in terms of x) are or

Note: since the degree of turns out to be 4, this means that there are really 4 solutions. So what happened? The twist is that each solution is counted twice. You'll most likely hear that these solutions are of multiplicity 2.
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