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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2
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3answers
428 views

Show that $\gcd(a + b, a^2 + b^2) = 1\mbox{ or } 2$ [duplicate]

How to show that $\gcd(a + b, a^2 + b^2) = 1\mbox{ or } 2$ for coprime $a$ and $b$? I know the fact that $\gcd(a,b)=1$ implies $\gcd(a,b^2)=1$ and $\gcd(a^2,b)=1$, but how do I apply this to that?
2
votes
3answers
496 views

How do you know when you can substitute certain limits into others?

I know there are some limits where you can't to certain substitutions such as $\sin(x)=x$ as $x$ approaches $0$. How do you know when you can or can't do that? I wish I could give you an example ...
2
votes
1answer
122 views

Is the square root of $4$ only $+2$? [duplicate]

Why is $4^{1/2}=+2$? It should also be $-2$ since both squared just give two only. Also why do we always represent root of $x$ on the right side of the number line?
2
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3answers
145 views

A tricky logarithms problem?

$ \log_{4n} 40 \sqrt{3} \ = \ \log_{3n} 45$. Find $n^3$. Any hints? Thanks!
2
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4answers
2k views

Solve the equation $\sqrt{3x-2} +2-x=0$

Solve the equation: $$\sqrt{3x-2} +2-x=0$$ I squared both equations $$(\sqrt{3x-2})^2 (+2-x)^2= 0$$ I got $$3x-2 + 4 -4x + x^2$$ I then combined like terms $x^2 -1x +2$ However, that can not ...
2
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1answer
194 views

Taylor polynomial approximation

How do you determine if adding more terms to the Taylor polynomial will improve its approximation of $f(p)$ or in other words, how do you determine if a Taylor series converges for a particular value ...
2
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1answer
235 views

Given that $\sec \theta = k$, $|k| \geq 1$, and that $\theta$ is obtuse, express in terms of $k$: $\cos \theta$, and $\csc \theta$

Given that $\sec \theta = k$, $|k| \geq 1$, and that $\theta$ is obtuse, express in terms of $k$: $\cos \theta$, and $\csc \theta$ For $\cos \theta$ I get: $\frac{-1}{k}, (as \theta$ is obtuse, ...
2
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0answers
120 views

Imposing condition of specification of product of $n$ of imaginary numbers on coefficients of imaginary numbers

I asked the same question but with some fatal mistake that makes the question unanswerable - so I decided to delete it and start new. Connecting from The set of numbers that when multiplied do not ...
2
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4answers
302 views

Fast square roots

I need to compute the square roots of lots of numbers. The numbers increase monotonically by fixed step. For example, 1, 2, 3, ..., 1 000 000. What is the fastest way to do so? Is it possible somehow ...
2
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2answers
339 views

Rationalizing the denominator of $\frac {\sqrt{10}}{\sqrt{5} -2}$

I have the expression $$\frac {\sqrt{10}}{\sqrt{5} -2}$$ I can't figure out what to do from here, I can't seem to pull any numbers out of either of the square roots so it appears that it must remain ...
2
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1answer
515 views

Rearranging equations with sine

I am working on a program which will predict the tides, but have come across a problem when using the simplified harmonic method of tidal prediction, I understand the whole thing but cannot do the ...
2
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2answers
1k views

Expand $\ln\left[\frac{(4x^5-x-1)\sqrt{x-7}}{(x^2+1)^3}\right]$.

Expand this expression to the greatest possible terms with the lowest possible exponents. $\ln\left[\dfrac{(4x^5-x-1)\sqrt{x-7}}{(x^2+1)^3}\right]$ There are two ways at which I approached this ...
2
votes
1answer
44 views

How do I transform the equation based on this condition?

If a and b are the roots of the equation $$2x^2-px+7=0$$ Then a-b is a root of ?
2
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1answer
538 views

Composition of two polynomials

How's to make the composition of two polynomials? According to this page: If $ P = (x^3 + x) $, $ Q = (x^2 + 1) $ then, $ P\circ Q = P\circ (x^2 + 1) = (x^2 + 1)^3 + (x^2 + 1) = x^6 + 3 x^4 + 4 x^2 ...
2
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2answers
631 views

Finding the remainder from equations.

I am having problems solving this question : When n is divide by 4 the remainder is 2 what will the remainder be when 6n is divided by 4 ? Ans=$0$ Here is what I have got so far ...
2
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3answers
800 views

Solve an absolute value equation simultaneously

My question is : Solve simultaneously $$\left\{\begin{align*}&|x-1|-|y-2|=1\\&y = 3-|x-1|\end{align*}\right.$$ What I did : $y=3 - |x-1|$ is given. Thus $y = 3-(x-1)$ or $y = ...
2
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1answer
68 views

$n^s=(n)_s+f(s)$, what is $f(s)$?

In the following equation, $$n^s=(n)_s+f(s)$$ What is general form for $f(s)$? Understand that, $$(n)_s=n(n-1)(n-2)\cdots(n-[s-1])=\text{ The Falling Factorial }$$ I have experimented with this ...
2
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3answers
617 views

How to fill up the gap between a typical advanced undergraduate algebraic curve course and High school basic geometry/precalculus course?

Based on this question i asked recently: A question about geometry of plane curve books, i think it is too advance for a HS student/ typical second or third year undergraduate math majors to read on ...
2
votes
1answer
173 views

Find all reals $a, b$ for which $a^b$ is also real

The title is pretty much clear, but here is a more precise formulation: Find all pairs $(a,b)\in\mathbb{R^2}$ for which $a^b$ is also real. I used a CAS to solve the problem and it says that the ...
2
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1answer
5k views

Learning how to flip equations

I took Algebra and Geometry in high school, never thought I'd use them, then became a programmer. I guess I was wrong. To date, I have the hardest time taking equations and "flipping them," ie: ...
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5answers
1k views

How do you divide a polynomial by a binomial of the form $ax^2+b$, where $a$ and $b$ are greater than one?

I came across a question that asked me to divide $-2x^3+4x^2-3x+5$ by $4x^2+5$. Can anyone help me?
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5answers
1k views

Need help finding smallest value of $x^2 + y^2$

I need to find the smallest value of $x^2 + y^2$ with the restriction $2x + 3y = 6$. This chapter focuses on the vertex formula.
2
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3answers
357 views

Finding an $f(x)$ that satisfies $f(f(x)) = 4 - 3x$

I need to find $f(f(x)) = 4 - 3x$ In other examples, such as $f(2)$, I can see that the result equates to $-2$ or $f(x^2)$ becomes $-3x^2 + 4$. Do I really just substitute $f(x)$ for $x$ and ...
2
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2answers
126 views

$\log_{12} 2=m$ what's $\log_6 16$ in function of $m$?

Given $\log_{12} 2=m$ what's $\log_6 16$ in function of $m$? $\log_6 16 = \dfrac{\log_{12} 16}{\log_{12} 6}$ $\dfrac{\log_{12} 2^4}{\log_{12} 6}$ $\dfrac{4\log_{12} 2}{\log_{12} 6}$ ...
2
votes
1answer
145 views

Solving inequalities comparing $f(x)$ to $0$ where $f$ is an elementary function

Any inequality comparing elementary functions can be rearranged to compare some elementary function $f$ to $0$. What is the best way to approach, in general, solving such inequalities at the ...
2
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2answers
1k views

Converting polar equation to cartesian coordinate polar equation and back again?

OK, so I have the following polar equation: $r = Θ/20$ And I would like to translate this a little to the right, and down from the polar origin. Now, I figure since I know cartesian coordinate ...
2
votes
2answers
398 views

Find the values of $m$ in the 2nd degree equation $mx^2-2(m-1)x-m-1=0$ so that it has one root between $-1$ and $2$

Find the values of $m$ in the 2nd degree equation $mx^2-2(m-1)x-m-1=0$ so that it has only one root between $-1$ and $2$. Like in this almost identical question there are two ways to solve this, one ...
2
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1answer
358 views

On Profit and loss

A dishonest dealer marks his goods $20\%$ above the cost price. He gives a discount of $10\%$ to the customer on the marked price and makes a profit by using a false weight of $900$ gms in place of ...
2
votes
4answers
171 views

How to sum up this series?

How to sum up this series : $$2C_o + \frac{2^2}{2}C_1 + \frac{2^3}{3}C_2 + \cdots + \frac{2^{n+1}}{n+1}C_n$$ Any hint that will lead me to the correct solution will be highly appreciated. EDIT: ...
2
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6answers
2k views

Cartesian Equation for the perpendicular bisector of a line

Find the Cartesian equation for the perpendicular bisector of the line joining A(2,3) and B(0,6) How do I do this? Thank you!
1
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0answers
34 views

Prove the identity $\tanh(N\textrm{acosh}\;a) = \vert \frac{g^{2N}-1}{g^{2N}+1}\vert$

During my recent study, I found an Identity which is of the form $$ \tanh(N\textrm{acosh}\;a) = \left\vert \frac{g^{2N}-1}{g^{2N}+1}\right\vert $$ where $a\geq1$ and $g>0$ satisfy ...
1
vote
2answers
52 views

How can I prove that the follow polynomial is irreducible in $\mathbb{Q}$?

How can I prove that $x^5 + 7x^4 + 2x^3 + 6x^2 - x + 8$ is irrudicible in $\mathbb{Q}$? I can't use the Eisenstein's criterion and I tryed to put this polynomial in $\mathbb{Z}_3$ and $\mathbb{Z}_5$. ...
1
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1answer
58 views

Help with $\lim_{x, y\to(0, 0)} \frac{x^2y}{x^4+y^2}?$ [duplicate]

The question asks to evaluate the limit and discuss the continuity of the function. I think I made steps in the right direction, but I'm not sure how to go from there $$\lim_{x, y\to(0, 0)} ...
1
vote
2answers
89 views

Ways of coloring the $7\times1$ grid (with three colors)

Hints only please! A $7 \times 1$ board is completely covered by $m \times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the ...
1
vote
1answer
68 views

What is the proof for this sum of sum generalized harmonic number?

I believe this sum: $$\sum_{m=2}^k\sum_{n=1}^{m-1}(nm)^{-s}$$ to be equal to $$\frac 12((H_k^{s})^2-H_k^{(2s)})$$ where $H_k^{s}$ is the generalized harmonic number. I only discovered this by ...
1
vote
3answers
96 views

Mid '|' in math?

What does this equation mean? What does the $|$ mean? $446617991732222310 | mn(m^k - n^k)$ Here is the complete question for reference - What is the smallest positive integer $k$, such that for ...
1
vote
3answers
76 views

Let $\theta=\frac{2 \pi}{67}$ consider the rotation matrix $A$. What is $A^{2010}$?

Let $\theta=\frac{2 \pi}{67}$. Consider the matrix $$A = \begin{pmatrix} \cos\theta & \sin\theta\\ -\sin \theta& \cos \theta \end{pmatrix} $$ Then the matrix $A^{2010}$ is? My ...
1
vote
0answers
239 views

Crossed Ladders Problem

Two ladders, one 10 meters long and the other 8 meters [long], have been placed in a trench as indicated in the opposite figure. Their point of intersection, M, is 3 meters from the base of the ...
1
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1answer
81 views

Polynomials, prove exercise question about question

There is a polynomial P with integer coefficients and with pairwise different integers $a,b,c$ . Prove that it is not possible for $P(a) = b$, $P(b)=c$, $P(c) = a$ First off I don't understand ...
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vote
1answer
107 views

Partial fraction decomposition on: $\frac{1}{(1-u^2)^2}$?

how does one perform Partial fraction decomposition on: $\large \frac{1}{(1-u^2)^2}$ ? the square at the denominator makes it a bit non-standard...
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1answer
73 views

How do I find the horizontal asymptote of $f(x)=\frac{\sin (x) }{x}$?

I can instantly see that there will be a vertical asymptote at $x=0$, however I am finding it quite a challenge to find a horizontal asymptote. I've drawn the graph and it seems as if the amplitude of ...
1
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4answers
133 views

Solving an equation over the reals: $ x^3 + 1 = 2\sqrt[3]{{2x - 1}}$

Solve the following equation over the reals:$$ x^3 + 1 = 2\sqrt[3]{{2x - 1}} $$ I noticed that 1 is a trivial solution, then I tried raising the equation to the 3rd, then dividing the polynomial by ...
1
vote
2answers
510 views

A tricky running problem

I'm having trouble with the following problem: Tom starts running towards a park which is at $800$m from him at speed $20$ m/s. Kate who starts running with Tom at $25$ m/s goes back and ...
1
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1answer
95 views

How to solve infinite series $\sum_{n=0}^\infty\frac{n}{2^{(n+1)}}$? [duplicate]

Can anyone please help me solve an infinite series: $$\sum_{n=0}^{\infty} \frac{n}{2^{(n+1)}}$$ I put it in Wolfram Alpha and got the result that it converges to $1$ I know that the infinite ...
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4answers
347 views

Finding the range and domain of $f(x)=\tan (x)$

I am attempting to find the range and domain of $f(x)=\tan(x)$ and show why this is the case. I can seem to find the domain relatively well, however I run into problems with the range. Here's what I ...
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3answers
1k views

How can I find the following product? $ \tan 20^\circ \cdot \tan 40^\circ \cdot \tan 80^\circ.$

How can I find the following product using elementary trigonometry? $$ \tan 20^\circ \cdot \tan 40^\circ \cdot \tan 80^\circ.$$ I have tried using a substitution, but nothing has worked.
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0answers
65 views

The elegant expression in terms of gcd and lcm - algebra - (2)

Definition: suppose a quantity $P$ is identified by $$ \frac{P}{k}\simeq \frac{P}{k}+1 $$ what we mean is that $$ P= 0\pmod{k}. $$ That means that when $P \to P+k$, then $$ \frac{P}{k}\to ...
1
vote
6answers
342 views

Prove that $\sum_{k=0}^n k^2{n \choose k} = {(n+n^2)2^{n-2}}$

Prove that: $$\sum_{k=0}^n k^2{n \choose k} = {(n+n^2)2^{n-2}}$$ i know that: $$\sum_{k=0}^n {n \choose k} = {2^n}$$ how to get the (n + n^2)?
1
vote
3answers
349 views

Prove that $\sec^2{\theta}=(4xy)/(x+y)^2$ only when $x=y$

Show that the equation below is only possible when $x=y$ $$ \sec^2{\theta}=\frac{4xy}{(x+y)^2}$$ The only way I can think of doing this is by rewriting it as $$ ...
1
vote
4answers
408 views

Finding the Limit in: $\lim\limits_{x\rightarrow1}\frac{\frac{1}{\sqrt{x}}-1}{x-1}$

Need some help finding this limit: $$\lim_{x\rightarrow1}\frac{\frac{1}{\sqrt{x}}-1}{x-1}$$ Here is what I have so far: $$\lim_{x\rightarrow1}\dfrac{\dfrac{1-\sqrt{x}}{\sqrt{x}}}{x-1}$$ ...