Solving 1995 Putnam Math Problem: Unclear Statement

[ehrenfest]

Homework Statement

Here is the problem:
http://www.unl.edu/amc/a-activities/a7-problems/putnam/-pdf/1995.pdf

Here is the solution:
http://www.unl.edu/amc/a-activities/a7-problems/putnam/-pdf/1995s.pdf

In the solution, I am not sure why the sentence that starts (ironically) with "Clearly, then" is true?

Homework Equations

The Attempt at a Solution

 

[morphism]

They probably mean (b^2 - 2) instead of (b^2 + 2), in which case it follows from the above observation.

 

[ehrenfest]

But the "above observation" contains a quartic polynomial and in that sentence it is a quadratic. I do not understand how that transformation happened...

 

[morphism]

Right, but do you notice anything special about the above quartic? Try setting y=x^2. Do you see the relevance of the phrase "positive square roots" now?

 

[ehrenfest]

I know I am missing something really obvious, but if you set y = x^2, you get

y^2 - (b^2-1)y+1 = 0

I am not sure where the sqrt of the linear coefficient comes from...

 

[ehrenfest]

I know I am missing something really obvious, but if you set y = x^2, you get

y^2 - (b^2-1)y+1 = 0

I am not sure where the sqrt of the linear coefficient comes from...

I see. The square root of the linear coefficient should not be there! Also, when they say positive square roots they mean the squares of the positive roots, right?

 

[ehrenfest]

Am I right about the square root being incorrect on the linear coefficient?