Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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16
votes
3answers
126 views

Intriguing Indefinite Integral: $\int ( \frac{x^2-3x+\frac{1}{3}}{x^3-x+1})^2 \mathrm{d}x$

Evaluate $$\int \left( \frac{x^2-3x+\frac{1}{3}}{x^3-x+1}\right)^2 \mathrm{d}x$$ I tried using partial fractions but the denominator doesn't factor out nicely. I also substituted ...
16
votes
2answers
402 views

Integer values of $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$?

What are the possible integer values of $$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$$ where $x$, $y$, and $z$ are positive integers? My suspicion is the the only integer values are $3$ and $5$, the former ...
15
votes
7answers
841 views

How to calculate $z^4 + \frac1{z^4}$ if $z^2 + z + 1 = 0$?

Given that $z^2 + z + 1 = 0$ where $z$ is a complex number, how do I proceed in calculating $z^4 + \dfrac1{z^4}$? Calculating the complex roots and then the result could be an answer I suppose, but ...
15
votes
12answers
2k views

How to prove $(a-b)^3 + (b-c)^3 + (c-a)^3 -3(a-b)(b-c)(c-a) = 0$ without calculations

I read somewhere that I can prove this identity below with abstract algebra in a simpler and faster way without any calculations, is that true or am I wrong? $$(a-b)^3 + (b-c)^3 + (c-a)^3 ...
15
votes
6answers
1k views

Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$.

Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$. I do not understand how to go about completing this problem or even where to start.
15
votes
4answers
750 views

Calculating $\sqrt{28\cdot 29 \cdot 30\cdot 31+1}$

Is it possible to calculate $\sqrt{28 \cdot 29 \cdot 30 \cdot 31 +1}$ without any kind of electronic aid? I tried to factor it using equations like $(x+y)^2=x^2+2xy+y^2$ but it didn't work.
15
votes
4answers
774 views

Proving that $\sum\limits_{i=1}^k i! \ne n^2$ for any $n$ [duplicate]

Possible Duplicate: How to prove that the number 1!+2!+3!+…+n! is never square? Show that $\displaystyle\sum\limits_{i=1}^k i!$ is never a perfect square for $k\ge4$ I could prove ...
15
votes
6answers
747 views

Solution to $1-f(x) = f(-x)$

Can we find $f(x)$ given that $1-f(x) = f(-x)$ for all real $x$? I start by rearranging to: $f(-x) + f(x) = 1$. I can find an example such as $f(x) = |x|$ that works for some values of $x$, but not ...
15
votes
6answers
3k views

No radical in the denominator — why? [duplicate]

Why do all school algebra texts define simplest form for expressions with radicals to not allow a radical in the denominator. For the classic example, $1/\sqrt{3}$ needs to be "simplified" to ...
15
votes
4answers
803 views

Factor $x^4 - 11x^2y^2 + y^4$

This is an exercise from Schaum's Outline of Precalculus. It doesn't give a worked solution, just the answer. The question is: Factor $x^4 - 11x^2y^2 + y^4$ The answer is: $(x^2 - 3xy -y^2)(x^2 + ...
15
votes
4answers
417 views

If $x$ and $y$ are rational numbers and $x^5+y^5=2x^2y^2,$ then $1-xy$ is a perfect square.

Prove that if $x, y$ are rational numbers and $$ x^5 +y^5 = 2x^2y^2$$ then $1-xy$ is a perfect square.
15
votes
4answers
1k views

How many solutions for this equation?

$$ \frac{x-4}{(x-1)} = \frac{1-4}{(x-1)} $$ Can someone tell me how many solutions are there for the above equation? MY APPROACH: I cross multiplied the equations and re-arranged to get a quadratic ...
15
votes
2answers
610 views

Intuitive ways to get formula of cubic sum

Is there an intuitive way to get cubic sum? From this post: combination of quadratic and cubic series and Wikipedia: Faulhaber formula, I get $$1^3 + 2^3 + \dots + n^3 = \frac{n^2(n+1)^2}{4}$$ I think ...
15
votes
3answers
2k views

Riddle with Pi = 3

This is a riddle someone posted on Google+, so please forgive it's triviality - I'm asking here because I just can't figure out what exactly is wrong, and it really bugs me ;) I think something is ...
15
votes
2answers
2k views

Significance of $\displaystyle\sqrt[n]{a^n} $?

There is a formula given in my module: $$ \sqrt[n]{a^n} = a \text{ if $n$ is odd } $$ $$ \sqrt[n]{a^n} = |a| \text{ if $n$ is even } $$ I don't really understand the differences between them, ...
15
votes
3answers
287 views

How can I prove $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}…}}}}=2$ [duplicate]

How can I prove $$\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}...}}}}=2$$ I don't know which method can be used for this?
15
votes
5answers
8k views

Good book for high school algebra

I'm gonna take a Calculus course next year, my professor suggest me to review high school algebra. I want to know, which book is good for refresh knowledge on high school algebra?
15
votes
2answers
10k views

How to prove a limit exists using the $\epsilon$-$\delta$ definition of a limit

I understand how to find a limit. I understand the concept of the $\epsilon$-$\delta$ definition of a limit. Can you walk me through what we're doing in this worked example? It is from my student ...
15
votes
1answer
1k views

Nested Square Roots

How would one go about computing the value of $X$, where $X=5^0+ \sqrt{5^1+\sqrt{5^2+\sqrt{5^4+\sqrt{5^8+\sqrt{5^{16}+\sqrt{5^{32}+\dots}}}}}}$ I have tried the standard way of squaring then trying ...
15
votes
2answers
3k views

Why does the discriminant of a cubic polynomial being less than 0 indicate complex roots?

The discriminant $\Delta = 18abcd - 4b^3d + b^2 c^2 - 4ac^3 - 27a^2d^2$ of the cubic polynomial $ax^3 + bx^2 + cx+ d$ indicates not only if there are repeated roots when $\Delta$ vanishes, but also ...
15
votes
1answer
315 views

Solve $x^x=2x$ where $x\in\mathbb C$.

Solve $x^x=2x$ where $x\in\mathbb C$. Obviously, one solution is $x=2$. By WA, another solution is $x=0.346...$. How to solve it analytically, e.g. using Lambert W function? Thank you.
15
votes
2answers
265 views

How much math do we need to prove all simple numeric identities?

Consider real numeric expressions build only from integers, operators $+,-,\times,/$ and taking a positive expression to a power (no variables involved), e.g. $$\frac{2}{7},\ 2^{1/2},\ ...
14
votes
7answers
700 views

$n^5-n$ is divisible by $10$?

I was trying to prove this, and I realized that this is essentially a statement that $n^5$ has the same last digit as $n$, and to prove this it is sufficient to calculate $n^5$ for $0-9$ and see that ...
14
votes
3answers
397 views

$ \tan 1^\circ \cdot \tan 2^\circ \cdot \tan 3^\circ \cdots \tan 89^\circ$

How can I find the following product using elementary trigonometry? $$ \tan 1^\circ \cdot \tan 2^\circ \cdot \tan 3^\circ \cdots \tan 89^\circ.$$ I have tried using a substitution, but nothing ...
14
votes
5answers
3k views

Inequality: $(x + y + z)^3 \geq 27 xyz$

Edit: $a,b,c$ and $x,y,z$ are positive, real numbers. Since $(a-b)^2 \geq 0~$, $a^2 + b^2 - 2ab\geq0~$ and $a^2 + b^2 \geq 2ab~$. Similarly, $a^2 + c^2 \geq 2ac~$ and $b^2 + c^2 \geq 2bc~$. ...
14
votes
3answers
108k views

How to Determine if a Function is One-to-One

I am looking for the "best" way to determine whether a function is one-to-one, either algebraically or with calculus. I know a common, yet arguably unreliable method for determining this answer would ...
14
votes
4answers
566 views

How to prove $\sum\limits_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$?

Other than the general inductive method,how could we show that $$\sum_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$$ Apart from induction,I tried with Wolfram Alpha to check the validity,but ...
14
votes
4answers
617 views

Process to show that $\sqrt 2+\sqrt[3] 3$ is irrational

How can I prove that the sum $\sqrt 2+\sqrt[3] 3$ is an irrational number ??
14
votes
3answers
556 views

An incorrect method to sum the first $n$ squares which nevertheless works

Start with the identity $\sum_{i=1}^n i^3 = \left( \sum_{i = 1}^n i \right)^2 = \left(\frac{n(n+1)}{2}\right)^2$. Differentiate the left-most term with respect to $i$ to get $\frac{d}{di} ...
14
votes
4answers
333 views

Show the identity $\frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a}=-\frac{a-b}{a+b}\cdot\frac{b-c}{b+c}\cdot\frac{c-a}{c+a}$

I was solving an exercise, so I realized that the one easiest way to do it is using a "weird", but nice identity below. I've tried to found out it on internet but I've founded nothingness, and I ...
14
votes
2answers
2k views

What is the best way to solve an equation involving multiple absolute values?

An absolute value expression such as $|ax-b|$ can be rewritten in two cases as $|ax-b|=\begin{cases} ax-b & \text{ if } x\ge \frac{b}{a} \\ b-ax & \text{ if } x< \frac{b}{a} \end{cases}$, ...
14
votes
2answers
371 views

how to compare $\sin(19^{2013}) $ and $\cos(19^{2013})$

how to compare $ \sin(19^{2013})$ and $\cos (19^{2013})$ or even find their value range with normal calculator? I can take $2\pi k= 19^{2013} \to \ln(k)= 2013 \ln(19)- \ln(2 \pi)=5925.32 \to k= ...
14
votes
8answers
27k views

Calculus book recommendations (for complete beginner)

Well I have not started calculus yet but I am really keen to. I would love if you suggest some books. Points to be noted: I really don't like the way textbooks are written so please no "textbooks" ...
14
votes
3answers
178 views

Finding the real solutions to $16^{x^{2} + y } + 16^{y^{2}+ x} = 1$

We have , $16^{x^{2} + y } + 16^{y^{2}+ x} = 1$ , then we have to find all the real values of $x$ and $y$.I tried this question but i am not able to proceed because I am not able to simplify this ...
14
votes
1answer
388 views

Why does Group Theory not come in here?

Here is a list of questions that I find quite similar, for the one and only reason that they all show much "symmetry". Which is at the same time my problem, because I don't have a very precise notion ...
14
votes
2answers
5k views

Relearning from the basics to Calculus and beyond.

Assume someone has very limited knowledge of math. (low level high school, 5-6 years ago) How would they learn from the basics of algebra, geometry and trigonometry to a solid foundation for calculus ...
14
votes
1answer
3k views

Using Vieta's theorem for cubic equations to derive the cubic discriminant

Background: Vieta's Theorem for cubic equations says that if a cubic equation $x^3 + px^2 + qx + r = 0$ has three different roots $x_1, x_2, x_3$, then $$\begin{eqnarray*} -p &=& x_1 + x_2 ...
14
votes
4answers
170 views

Proving $\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)<3$ for all positive integers $n$

Prove that $\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)<3$ for all positive integers $n$ This problem is copied from Math Olympiad Treasures ...
14
votes
4answers
787 views

How find the value of the $x+y$

Question: let $x,y\in \Bbb R $, and such $$\begin{cases} 3x^3+4y^3=7\\ 4x^4+3y^4=16 \end{cases}$$ Find the $x+y$ This problem is from china some BBS My idea: since ...
14
votes
1answer
152 views

How to prove that the problem cannot be solved by the four Arithmetic Operations?

The original prolbem is as in the figure: Suppose the square has unit side length, find the area of blue region. The exact solution is: $$\begin{aligned}S=&\frac{\pi-\sqrt{7}}{4}+2 ...
14
votes
2answers
205 views

Prove $\log_5{30}<\log_8{81}$

It's easy to prove this by calculator or computer, and I wonder can we prove that $$\log_5{30}<\log_8{81}\tag 1$$ by pencil and paper ? Thanks in advance ! Edit: $(1)$ can be written as ...
14
votes
4answers
145 views

High School Advanced Functions: Clarifying log rules in a log equation - $\log(x^2) = 2$, Solve for x.

I got in an argument with my teacher for the possible solutions of x. From some sources i found that because x is squared, negative values should be possible; however, my teacher insists that: $$ ...
14
votes
1answer
696 views

Expressing the maximum of several variables using elementary functions [duplicate]

It's well-known that $$\max(a,b)=\frac{a+b+|a-b|}{2}.$$ Is there a (good) generalization to several variables? Of course $\max(a,b,c)=\max(a,\max(b,c))$ and so ...
13
votes
10answers
4k views

Taking Calculus in a few days and I still don't know how to factorize quadratics

Taking Calculus in a few days and I still don't know how to factorize quadratics with a coefficient in front of the 'x' term. I just don't understand any explanation. My teacher gave up and said just ...
13
votes
6answers
1k views

If $2^x=0$, find $x$.

If $2^x=0$, find $x$. Solution: I know range of $2^x$ function is $(0,\infty)$. So $2^x=0$ is not possible for any real value of $x$ Hence, equation is wrong. We can't find value of $x$. Am I ...
13
votes
3answers
521 views

Is there any “superlogarithm” or something to solve $x^x$? [duplicate]

Is there any "superlogarithm" or something to solve an equation like this: $$x^x = 10?$$
13
votes
4answers
814 views

Prove that $x = 2$ is the unique solution to $3^x + 4^x = 5^x$ where $x \in \mathbb{R}$ [duplicate]

Yesterday, my uncle asked me this question: Prove that $x = 2$ is the unique solution to $3^x + 4^x = 5^x$ where $x \in \mathbb{R}$. How can we do this? Note that this is not a diophantine ...
13
votes
3answers
1k views

Simplification of expressions containing radicals

As an example, consider the polynomial $f(x) = x^3 + x - 2 = (x - 1)(x^2 + x + 2)$ which clearly has a root $x = 1$. But we can also find the roots using Cardano's method, which leads to $$x = ...
13
votes
4answers
827 views

Solving for the implicit function $f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right)$ and $f(1)=1$

How can I find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(1)=1$ and $$f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right)$$ for all real numbers $x$ and $y$ with $y\neq0$? PS. This is ...
13
votes
3answers
632 views

Evaluate $ \displaystyle \lim_{x\to 0}\Bigg( \frac {(\cos(x))^{\sin(x)} - \sqrt{1 - x^3}}{x^6}\Bigg) $

Evaluate $$ \displaystyle \lim_{x\to 0}\Bigg( \frac {(\cos(x))^{\sin(x)} - \sqrt{1 - x^3}}{x^6}\Bigg) $$ I tried to use L'Hospital rule but it got very messy. Moreover I also tried to analyze ...