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- Feb 14, 2012

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Find real solution(s) to the equation \(\displaystyle (x^2-9x-1)^{10}+99x^{10}=10x^9(x^2-1)\)

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- Feb 14, 2012

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Find real solution(s) to the equation \(\displaystyle (x^2-9x-1)^{10}+99x^{10}=10x^9(x^2-1)\)

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- Feb 7, 2012

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Hint 1:Find real solution(s) to the equation \(\displaystyle (x^2-9x-1)^{10}+99x^{10}=10x^9(x^2-1)\)

Divide both sides by $x^{10}$.

Let $y = x-\frac1x$.

What happens when $y=10$?

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- Feb 14, 2012

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I thank you for taking the time to share your helpful hints in this challenge problem.

My approach is different from yours and I wish to share it here too.

Hint 1:

We first rewrite the equation as \(\displaystyle (x^2-9x-1)^{10}-10x^9(x^2-9x-1)+9x^{10}=0\)

Hint 2:

Assume that \(\displaystyle (x^2-9x-1)^{10}+9x^{10}=10x^9(x^2-9x-1)\) is true.

Hint 3:

Hint 4:

The original problem is solved if we solve for x for \(\displaystyle x^2-10x-1=0\)

I failed to realize that there are two other real solutions to this problem. Would you mind to teach me how to tell how many real solutions there are for this particular problem?That will give you two real solutions. There are two others, but I do not think that they can be explicitly determined.

Thanks in advance.

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- Feb 7, 2012

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That was my silly mistake. There are no other real solutions. Sorry about that.I failed to realize that there are two other real solutions to this problem. Would you mind to teach me how to tell how many real solutions there are for this particular problem?

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- Feb 14, 2012

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I see...hey, don't worry about that!That was my silly mistake. There are no other real solutions. Sorry about that.

You are and always will be one of my favorite mathematicians at this site!

- Aug 30, 2012

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Graph it!! Hey, I'm a Physicist. (Physics isFind real solution(s) to the equation \(\displaystyle (x^2-9x-1)^{10}+99x^{10}=10x^9(x^2-1)\)

-Dan

- Jan 26, 2012

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Indeed. I am now confident the real roots are between $-\infty$ and $+\infty$, more or lessGraph it!! Hey, I'm a Physicist. (Physics isalwaysbetter with beer.)

-Dan

- Feb 13, 2012

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Unfortunately in this case graphing in not a comfortable way to arrive to the solution ...Graph it!! Hey, I'm a Physicist. (Physics isalwaysbetter with beer.)

-Dan

All is

Kind regards

$\chi$ $\sigma$