Find the coefficient of $x^{-2}$ in $$(x-1/x)^{12}$$   Can anyone help me with this by providing the process to find the answer? I know the answer is $-792$ but I can't find it.
121183
Find the coefficient of the term $x^2$ in $$\left(x+\frac 2x\right)^{4}$$   This is what I have done so far.  We know $a$ = $x$ and $b$ = $\frac2x$ and $n$ = 4  So what I have done is $4c2$ x $(x)^2$ x ($\frac2x$)$^2$  As we know that $^2$ + $^2$ = 4, and 4 is the $n$ value.  The answer is one though?  What have I done wrong?   P.S.  was the tutorial that helped me.
41472
How can I find the inverse function of $f(x) = x^x$? I cannot seem to find the inverse of this function, or any function in which there is both an $x$ in the exponent as well as the base. I have tried using logs, differentiating, etc, etc, but to no avail.
21269
Can someone give me a hint on how can I solve the equation $$x^3 - x - 1 =0?$$  Thank you!
65496
Given that $A$ is a constant coefficient matrix, how would one find a non-zero solution to the following equation: $$ x''+2Ax'+A^2x=0 $$ I am thinking this could incorporate systems of linear equations, but I am not sure how to use $x'=Ax$ or $x'=P(t)x +g(t)$ in this situation.   Any hints or suggestions are appreciated.
110616
The sum of the series $$\binom{n}{0}-\binom{n-1}{1}+\binom{n-2}{2}-\binom{n-3}{3}+..........$$   $\bf{My\; Try::}$ We can write it as $\displaystyle \binom{n}{0} = $ Coefficient of $x^0$ in $(1+x)^n$   Similarly $\displaystyle \binom{n-1}{1} = $ Coefficient of $x^1$ in $(1+x)^{n-1}$  Similarly $\displaystyle \binom{n-2}{2} = $ Coefficient of $x^2$ in $(1+x)^{n-2}$  Now, how can I solve it after that, Help Required, Thanks
100726
The product of the two roots of $\sqrt{2014}x^{\log_{2014} x}=x^{2014}$ is an integer. Find its units digit.   I'm quite unable to solve the problem given. I have no idea how to deal with that $\sqrt{2014}$ term nor the logarithms in the exponent. Is it even necessary to solve for each individual root or is there some other way to progress through the problem?  I'm really just a novice so I'd appreciate if anyone could show me the process or processes, if multiple. Thank you very much.
90268
The question is, how many times does $f(C_r)$ go around $0$ for $C_r = \{x \in \mathbb{C}: |x|=r\}$ and a polynomial with complex coefficients $f$.    The answer is clear to me when $f(x) = ax^n + b$ but I don't see how exactly middle terms affect the result. How to see this without a graph? Can this be found for any $r$?    Many thanks
56554
I apologize if this is too obvious, but I crave for a reasonable definition/explanation - why is  $(-1)(-1)=1$?  Something akin to the reason $a^{-1}=\frac{1}{a}$, I.e. because $a^{-1}a^1=a^{1-1=0}=1$, so we get $a^{-1}=\frac{1}{a^1}=\frac{1}{a}$ by dividing both sides by $a^1$.  That sort of stuff.
16282
How to solve this equation $$x^{2}=2^{x}$$   where $x \in \mathbb{R}$.  Por tentativa erro consegui descobri que $2$ é uma solução, mas não encontrei um método pra isso. Alguma sugestão?(*)  (Translation: By trying different values I've found that $2$ is a solution, but I couldn't find any method to this though. Any suggestions? )
100189
Find the number of solutions to the equation $$2011^x+2012^x+2013^x-2014^x=0$$  The answer seems to be zero, but I have no idea why. Please avoid considering complex solutions and other scary things.
15348
Find all $x$ such that $$x^6=(x+1)^6.$$   So far, I have found the real solution $x= -\frac{1}{2}$, and the complex solution $x = -\sqrt[3]{-1}$.  Are there more, and if so what/how would be the most efficient find all the solutions to this problem? I am struggling to find the rest of the solutions.
45171
Show that there is at least one real value of $x$ for which      $$x^{1/3} + x^{1/2} = 1$$   I did draw the graphs of $x^{1/3}$ and $1-x^{1/2}$ and showed that they met at a point, but I don't think it's a good algebraic proof.  How should I proceed after substituting $x$ for $z^{1/6}$ and getting a polynomial in terms of $z$
122151
How do you approach a problem like (solve for $x$):  $$x^{x^{x^{x^{...}}}}=2$$  Also, I have no idea what to tag this as.   Thanks for any help.
124038
What is the coefficient of $x^{11}$ in the power series expansion of $\frac 1{1-x-x^4}$?    How do I do power series expansions?
68065
I saw this problem on a problem set and I have absolutely no idea how to proceed in a feasible way.     Does there exist a polynomial $f(x)$ with real coefficients such that $f(x)^2$ has fewer nonzero coefficients than $f(x)$?   Any help would be appreciated.
114427
The coefficient of $x^9$ in the expansion of $(1+x)(1+x^2)(1+x^3)...(1+x^{100})$.  I tried the following concept, how to sum 9 using 1-9 only without repetition.  1:9, 2:1+8,3:7+2, 4: 6+3, 5: 5+4,  6:1+2+6 ,7:1+3+5, 8:2+3+4  The answer is 8.  How will it be solved using Binomial theorem.
106183
Find the number of solutions of $x^2=x\mod m$ for any integer $m$.  I was thinking about bringing the $x$ over to the other side and somehow apply chinese remainder thm.