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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20
votes
1answer
272 views

Show that $x^{35}+\dfrac{20205}{2+x^{17}+\cos^2x}=100$ has no root $x\in \mathbb{R}$

Show that $x^{35}+\dfrac{20205}{2+x^{17}+\cos^2x}=100$ has no root $x\in \mathbb{R}$. By plotting graph I have seen that there are no roots for $x$. Can somebody prove it theoretically?
20
votes
5answers
353 views

How to solve $\sqrt {1+\sqrt {4+\sqrt {16+\sqrt {64+\sqrt {256\ldots }}}}}$

How to solve this equation? $$x=\sqrt {1+\sqrt {4+\sqrt {16+\sqrt {64+\sqrt {256\ldots }}}}}.$$ Answer: $x=2$
20
votes
1answer
3k views

What was Ramanujan's solution?

The wikipedia entry on Ramanujan contains the following passage: One of his remarkable capabilities was the rapid solution for problems. He was sharing a room with P. C. Mahalanobis who had a ...
19
votes
4answers
660 views

Calculate $\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$

I'm an eight-grader and I need help to answer this math problem. Problem: Calculate $$\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$$ This one is very hard for ...
19
votes
10answers
3k views

Why are equations written by equating something to zero?

A linear equation is $$ ax + b = 0 ; \,\, \,\, a\neq 0 $$ A quadratic equation is $$ax^2 + bx + c = 0 ; \,\, a\neq 0 $$ And so on... Why are all these equations written as $\dots = 0 $? Why do ...
19
votes
2answers
4k views

Integration by partial fractions; how and why does it work?

Could someone take me through the steps of decomposing $$\frac{2x^2+11x}{x^2+11x+30}$$ into partial fractions? More generally, how does one use partial fractions to compute integrals ...
19
votes
6answers
2k views

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

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.
19
votes
6answers
5k views

Can $x^3+3x^2+1=0$ be solved using high school methods?

I encountered the following problem in a high-school math text, which I wasn't able to solve using factorization/factor theorem: Solve $x^3+3x^2+1=0$ Am I missing something here, or is indeed a more ...
19
votes
8answers
3k views

Why do negative exponents work the way they do? [closed]

Why is a value with a negative exponent equal to the multiplicative inverse but with a positive exponent? $$a^{-b} = \frac{1}{a^b}$$
19
votes
9answers
6k views

System of nonlinear equations that leads to cubic equation

The system of equations are: $$\begin{align}2x + 3y &= 6 + 5x\\x^2 - 2y^2 - (3x/4y) + 6xy &= 60\end{align}$$ I can solve it through substitution but it is an arduous process to reach this ...
19
votes
6answers
2k views

Helping my daughter with her homework: solving an algebra word problem.

Three bags of apples and two bags of oranges weigh $32$ pounds. Four bags of apples and three bags of oranges weigh $44$ pounds. All bags of apples weigh the same. All bags of oranges weigh the ...
19
votes
4answers
1k views

Why dividing by zero still works

Today, I was at a class. There was a question: If $x = 2 +i$, find the value of $x^3 - 3x^2 + 2x - 1$. What my teacher did was this: $x = 2 + i \;\Rightarrow \; x - 2 = i \; \Rightarrow \; (x - ...
19
votes
7answers
2k views

Solving $\sqrt{x+5} = x - 1$

I'm currently learning about radicals and simplifying them, and I came across this problem on the internet and tried to solve it: $$\sqrt{x+5} = x - 1$$ So I used this logic: $$ \begin{align} ...
19
votes
5answers
521 views

Prove that: $ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$

How to prove the following trignometric identity? $$ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$$ Using half angle formulas, I am getting a number for $\cot7\frac12 ^\circ $, but I don't ...
19
votes
2answers
20k 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 ...
19
votes
3answers
4k views

How do you handle the floor and ceiling function in an equation?

I tried to do some math in a blog post of mine and came to one with a floor function. I wasn't sure how to deal with it so I just ignored it, and then added the ceiling function in my final equation ...
19
votes
4answers
382 views

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

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 ...
19
votes
4answers
464 views

An obvious pattern to $i\uparrow\uparrow n$ that is eluding us all?

Start with $i=\sqrt{-1}$. This will be $a_1$. $a_2$ will be $i^i$. $a_3$ will be $i^{i^{i}}$. $\vdots$ etc. In Knuth up-arrow notation: $$a_n=i\uparrow\uparrow n$$ And, amazingly, you can ...
19
votes
1answer
2k views

Averaging 2 roots of a cubic polynomial

Consider a cubic polynomial, $p(x)=k(x-a)(x-b)(x-c)$ where $k$ is some constant and $a,b,c$ its $3$ roots (not necessarily distinct, not necessarily real). It is very simple to show that if you ...
19
votes
4answers
226 views

Inclusion-exclusion-like fractional sum is positive?

Let $A_1,A_2,\ldots,A_n$ be finite nonempty sets. Is it true that $$\sum_{i=1}^n\frac{1}{|A_i|}-\sum_{1\leq i<j\leq n}\frac{1}{|A_i\cup A_j|}+\sum_{1\leq i<j<k\leq n}\frac{1}{|A_i\cup ...
18
votes
3answers
12k views

If there are $74$ heads and $196$ legs, how many horses and humans are there? [closed]

I was going through some problems then I arrived at this question which I couldn't solve. Does anyone know the answer to this question? One day, a person went to a horse racing area. Instead of ...
18
votes
8answers
1k views

Find $x$ such that $\sqrt{x+\sqrt{x+7}}\in \mathbb{N}$

Find $x$ such that $$\sqrt{x+\sqrt{x+7}}\in \mathbb{N}$$ I tried many ways: $$\sqrt{x+\sqrt{x+7}}=n$$ $$\sqrt{x+\sqrt{x+7}}^2=n^2$$ $$x+\sqrt{x+7}=n^2$$ then solve for $x$ but didn't do with ...
18
votes
10answers
778 views

hand evaluate $\sqrt{e}$

I have seen this question many times as a example of provoking creativity. I wonder how many ways are there to evaluate $\sqrt{e}$ as accurately as possible. The obvious way I can think of is to use ...
18
votes
3answers
169k 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 ...
18
votes
2answers
3k views

How to find the sum of this series : $1+\frac{1}{2}+ \frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{n}$

Problem : How to find the sum of this series : $1+\frac{1}{2}+ \frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{n}$ This is a Harmonic progression : So is this formula correct to sum the series : ...
18
votes
2answers
3k 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 $|ax-b|=\begin{cases} ax-b & \text{ if } x\ge \frac{b}{a} \\ b-ax & \text{ if } x< \frac{b}{a} \end{cases}$, ...
18
votes
2answers
36k views

Weird E letter? (sigma) [duplicate]

Possible Duplicate: What does the math notation $\sum$ mean? My school's prescribed book uses the weird letter E character without explaining what it is in the first chapter when it talks ...
18
votes
2answers
611 views

On the “funny” identity $\tfrac{1}{\sin(2\pi/7)} + \tfrac{1}{\sin(3\pi/7)} = \tfrac{1}{\sin(\pi/7)}$

This equality in the title is one answer in the MSE post Funny Identities. At first, I thought it had to do with $7$ being a Mersenne prime, but a little experimentation with Mathematica's integer ...
18
votes
1answer
9k views

Strange old multiplication table found in Oklahoma school

Today I read an article about chalk boards from 1917 discovered in an Oklahoma school. One of the chalkboards included the following curious image: (Oklahoma City Public Schools) The article ...
18
votes
2answers
813 views

High school algebra textbooks for gifted students

Cross-posted to Math Educators Stack Exchange. (link) I am looking for high school algebra/mathematics textbooks targeted at talented students, as preparation for fully rigorous calculus à la Spivak. ...
18
votes
2answers
418 views

Find $f(x)$ where $ f(x)+f\left(\frac{1-x}x\right)=x$

What function satisfies $ f(x)+f\left(\frac{1-x}x\right)=x$ ?
17
votes
4answers
3k views

Are the equations $2x - 2y = 11, x = y - 2$ unsolvable?

My 9th grade son had this math problem, which seemed unsolvable to me: $$2x - 2y = 11$$ $$x = y - 2$$ So we can use substitution to come up with: $$2(y - 2) - 2y = 11$$ Now distribute: $$2y - 4 ...
17
votes
3answers
2k views

Find all the integral solutions to $x^6-y^6+3x^4y-3y^4x+y^3+3x^2+3x+1=0$

If $x,y\in\mathbb{Z}_{+}$, then find all the integral solutions to: $$x^6-y^6+3x^4y-3y^4x+y^3+3x^2+3x+1=0$$ I tried solving this question for an hour but still couldn't get it. I tried ...
17
votes
10answers
1k views

Find the integer closest to $\ln(2013)$

I encounter such a problem, in a Maths contest, to find out the closest integer to $\ln(2013)$, without using a calculator. I really get stuck. I tried to turn $\ln(2013)$ into ...
17
votes
4answers
1k views

Not quite Fermat's Last Theorem

Prove that the equation $n^a + n^b = n^c$, with $a,b,c,n$ positive integers, has infinite solutions if $n=2$, and no solution if $n\ge3$.
17
votes
3answers
459 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 ...
17
votes
8answers
3k views

Kid's homework: 4 equations 5 unknowns? Going crazy!

I'm new here, and I'm hoping someone can help out. My 10 year old son has been set a maths problem, which I can't solve. I've got a PhD in neuroscience and do a fair amount of matlab stuff (data ...
17
votes
4answers
715 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 ...
17
votes
3answers
1k views

What skill do I lack to factor multivariate polynomials?

Ok so I can factor easily regular quadratic polynomials, i.e. $5x^2+7x+9$ (I'm not sure whether that's prime, just made it up), and I was working on solving $y^2+(x^2+2x−2)y+(x^3−x^2−2x)$ by ...
17
votes
3answers
25k views

How to calculate percentage of value inside arbitrary range?

So pardon if this is a simple question... I have a slider that returns a value in a given range, so: min: 174 max: 424 slider current value: 230 I want to treat ...
17
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, ...
17
votes
2answers
1k views

Interesting Question on Ants

A horizontal stick is one metre long. Fifty ants are placed in random positions on the stick, pointing in random directions. The ants crawl head first along the stick, moving at one metre per minute. ...
17
votes
4answers
1k views

Why do all parabolas have an axis of symmetry?

And if that's just part of the definition of a parabola, I guess my question becomes why is the graph of any quadratic a parabola? My attempt at explaining: The way I understand it after some ...
17
votes
3answers
561 views

When are algebraic expressions equivalent?

This question arose when I was going to determine the domain for $f \circ f(x)$. Let $f(x) = \dfrac{1-x}{1+x}$. $f \circ f(x) = x, \quad$ But the domain is not $\mathbb{R}$ because $f(x)$ is undefined ...
17
votes
2answers
436 views

Trig sum: $\tan ^21^\circ+\tan ^22^\circ+…+\tan^2 89^\circ = ?$

As the title suggests, I'm trying to find the sum $$\tan^21^\circ+\tan^2 2^\circ+...+\tan^2 89^\circ$$ I'm looking for a solution that doesn't involve complex numbers, or any other advanced branch in ...
17
votes
2answers
382 views

Rigorous proof that $\displaystyle\frac{1}{3} = 0.333\ldots$

I'm a PreCalculus student trying to find a rigorous proof that $\displaystyle\frac{1}{3} = 0.333\ldots$, but I couldn't find it. I think (just think) that this proof would start by proving that ...
17
votes
2answers
256 views

square root / factor problem $(A/B)^{13} - (B/A)^{13}$

Let $A=\sqrt{13+\sqrt{1}}+\sqrt{13+\sqrt{2}}+\sqrt{13+\sqrt{3}}+\cdots+\sqrt{13+\sqrt{168}}$ and $B=\sqrt{13-\sqrt{1}}+\sqrt{13-\sqrt{2}}+\sqrt{13-\sqrt{3}}+\cdots+\sqrt{13-\sqrt{168}}$. Evaluate ...
17
votes
1answer
115k views

Solving Triangles (finding missing sides/angles given 3 sides/angles)

What is a general procedure for "solving" a triangle—that is, for finding the unknown side lengths and angle measures given three side lengths and/or angle measures?
17
votes
1answer
263 views

Find all polynomials $\sum_{k=0}^na_kx^k$, where $a_k=\pm2$ or $a_k=\pm1$, and $0\leq k\leq n,1\leq n<\infty$, such that they have only real zeroes

Find all polynomials $\sum_{k=0}^na_kx^k$, where $a_k=\pm2$ or $a_k=\pm1$, and $0\leq k\leq n,1\leq n<\infty$, such that they have only real zeroes. I've been thinking about this question, but ...
16
votes
13answers
2k views

$xy=1 \implies $minimum $x+y=$?

If $x,y$ are real positive numbers such that $xy=1$, how can I find the minimum for $x+y$?