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
110 views

How do I isolate $y$ in $y = 4y + 9$?

$y = 4y + 9$ How do I isolate y? Can I do $y = 4y + 9$ $\frac{y}{4y} = 9$ etc Also some other questions please: $\frac{5x + 1}{3} - 4 = 5 - 7x$ In the above (1), if I want to remove the '3' ...
2
votes
2answers
48 views

Proving a logarithmic inequality

I'm interested why this is true: $$ \text{Considering }\forall (x,y,z) \in (1,\infty) $$ The following holds: $$\log_xy^z+\log_x{z^y}+log_y{z^x} \geq \frac{3}{2}$$ This is taken from a ...
2
votes
2answers
185 views

2011 AIME Problem 12, probability round table

Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate ...
2
votes
1answer
57 views

Finding $\prod_{k=0}^n(1+0.5^{2^k})$ [closed]

Find the product $$\prod_{k=0}^n(1+0.5^{2^k})$$ I tried but I couldn't, any help?
2
votes
2answers
108 views

simplify cos 1 degree + cos 3 degree +…+cos 43 degree?

I am currently working on a problem and reduced part of the equations down to $\cos(1^\circ)+\cos(3^\circ)+.....+\cos(39^\circ)+\cos(41^\circ)+\cos(43^\circ)$ How can I calculate this easily using ...
2
votes
2answers
87 views

$\frac{(2n)!}{4^n n!^2} = \frac{(2n-1)!!}{(2n)!!}=\prod_{k=1}^{n}\bigl(1-\frac{1}{2k}\bigr)$

i cant see why we have : $$\frac{(2n)!}{4^n n!^2} = \frac{(2n-1)!!}{(2n)!!}$$ $$\dfrac{(2n-1)!!}{(2n)!!} =\prod_{k=1}^{n}\left(1-\dfrac{1}{2k}\right),$$ Even i see the notion of ...
2
votes
1answer
128 views

sawtooth function.

Let $x \in\mathbb{R}$ $f(x)=\{ x\}$ be fractional part function or sawtooth function. and $F_x=\{f(n\ x);\ n\in\mathbb{N}^*\}$ Show that $x\in\mathbb{Q}\Longleftrightarrow \ \exists\ q ...
2
votes
2answers
100 views

Functional equation - Understading an easy step in my solution.

I am trying to solve the equation and find all $f: \mathbb{N} \rightarrow \mathbb{N}$ such that: $f(m+f(n))=f(f(m))+f(n)$ for all $n, m \in \mathbb{N_{0}} $. A reasonable approach to begin with ...
2
votes
3answers
59 views

Writing without negative exponent

can someone please describe to me the method behind writing numbers without negative exponents such as: Maybe just show me the logic / process? Especially for number (c) because fractions really ...
2
votes
1answer
114 views

Sum of all real solutions for $x$ to the equation $\displaystyle (x^2+4x+6)^{{(x^2+4x+6)}^{\left(x^2+4x+6\right)}}=2014.$

Find the sum of all real solutions for $x$ to the equation $\displaystyle (x^2+4x+6)^{{(x^2+4x+6)}^{\left(x^2+4x+6\right)}}=2014.$ $\bf{My\; Try::}$ Let $y=x^2+4x+6 = (x+2)^2+2\geq 2$. So our exp. ...
2
votes
2answers
95 views

On proving an identity given a system of trig equations

We are given the following: $$a^2 + b^2 + 2ab\cos\theta = 1 \tag1$$ $$d^2 + c^2 + 2cd\cos\theta = 1 \tag2$$ $$ac + bd + (ad + bc)\cos\theta = 0\tag3$$ It is required to prove that: $$a^2 + c^2 = ...
2
votes
2answers
142 views

derivation of simple linear regression parameters

I know there are some proof in the internet, but I attempted to proove the formulas for the intercept and the slope in simple linear regression using Least squares, some algebra, and partial ...
2
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2answers
633 views

How do I find the relationships between two variables in a formula that are on the same side of the equals sign?

When I've been asked to find the relationship between two variables in a formula (assuming the other variables are constant), it's generally been things like find the relationship between $F$ and $R$ ...
2
votes
2answers
503 views

Does Fermat's Little Theorem work on polynomials?

Let $p$ be a prime number. Then if $ f(x) = (1+x)^p$ and $g(x) = (1+x)$, then is $f \equiv g \mod p$? I'm trying to prove that for integers $a > b > 0$ and a prime integer $p$, ${pa\choose b} ...
2
votes
3answers
112 views

How prove this $\frac{a}{(bc-a^2)^2}+\frac{b}{(ca-b^2)^2}+\frac{c}{(ab-c^2)^2}=0$

let $a,b,c\in R$,if such $$\dfrac{1}{bc-a^2}+\dfrac{1}{ca-b^2}+\dfrac{1}{ab-c^2}=0$$ show that $$\dfrac{a}{(bc-a^2)^2}+\dfrac{b}{(ca-b^2)^2}+\dfrac{c}{(ab-c^2)^2}=0$$ this problem have nice ...
2
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2answers
75 views

Induction: $2^n = \sum_{v=0}^{n} \binom{n}{v}$ [duplicate]

I have to prove the following identity for $n \in \mathbb{N}$: $\displaystyle 2^n = \sum_{v=0}^{n} \binom{n}{v}$ Is there a way to show it through induction? Or is there a easier way? My steps so ...
2
votes
1answer
120 views

Is it true that $\forall b \forall c \forall x ((x^2 + bx + c \neq 0) \rightarrow b^2 - 4c < 0)$?

Well, I proved that $\forall b \forall c (b^2 - 4c \geq 0 \rightarrow \exists x(x^2 + bx + c = 0))$. This implies that $\forall b \forall c (\neg \exists x(x^2 + bx + c = 0) \rightarrow b^2 - 4c < ...
2
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3answers
533 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
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1answer
129 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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1answer
232 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
votes
1answer
133 views

Proof for an equality involving square roots

While trying to solve this problem, I stumbled upon the following equality $$ \sqrt{\sqrt{2x}+\sqrt{x+k}}+\sqrt{\sqrt{2x}+\sqrt{x-k}}=(\sqrt2+1) \left( ...
2
votes
1answer
6k views

What are the most famous (common used) precalculus books and its differences?

I'm trying to decide which one to pick up to begin a self study of mathematics. One of the factors is how much content is covered and the amount of associated solved problems the book has. EDIT: ...
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
votes
4answers
313 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
votes
1answer
76 views

Sequence $a_k=1-\frac{\lambda^2}{4a_{k-1}},\ k=2,3,\ldots,n$.

Consider the sequence $a_1, a_2,\ldots,a_n$ with $a_1=1$ and defined recursively by $$a_k=1-\frac{\lambda^2}{4a_{k-1}},\ k=2,3,\ldots,n.$$ Find $\lambda>1$ such that $a_n=0$. The answer is ...
2
votes
1answer
539 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
121 views

How can I find a general formula describing this piecewise function?

I have a set of scores $s \in[1;40]$, where $s$ is integer. I want to map each score to a index, like this: $$1 \le s \le 5 \to 0 \\ 6 \le s \le 10 \to 1 \\ 11 \le s \le 20 \to2\\ 21 \le s \le 30 ...
2
votes
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
votes
2answers
670 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
votes
1answer
189 views

Express each of the following expressions in the form $2^m3^na^rb^s$, where $m$, $n$,$ r$ and $ s$ are positive integers.

I just recently started relearning math as an adult, this should be easy but I have trouble understanding what the actual question is. I am not just looking for the answer to this, I merely wish to ...
2
votes
1answer
156 views

Is there easier way to calculate the limit of this function?

$$ \lim_{K\rightarrow\infty}\frac{(1-\epsilon)^K}{1+(1-\epsilon)^K}\frac{\sum_{i=1}^{\frac{K-1}{2}}\left(\begin{array}{l} K \\ i ...
2
votes
3answers
907 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
votes
1answer
6k 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: ...
2
votes
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?
2
votes
3answers
364 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
votes
2answers
129 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
972 views

Interesting problem on “neighbor fractions”

This is from I. M. Gelfand's Algebra book. Fractions $\displaystyle\frac{a}{b}$ and $\displaystyle\frac{c}{d}$ are called neighbor fractions if their difference $\displaystyle\frac{ad - bc}{bd}$ ...
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
419 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
votes
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
vote
2answers
101 views

Confusion regarding taking the square root given an absolute value condition.

From the Generating function for Legendre Polynomials: $$\Phi(x,h)=(1-2xh+h^2)^{-1/2}\quad\text{for}\quad \mid{h}\,\mid\,\lt 1$$ My text states that: For ...
1
vote
2answers
55 views

Determining whether there are solutions to the cubic polynomial equation $x^3 - x = k - k^3$ other than $x = -k$ for a given parameter $k$

Let $k$ be a real parameter, and consider the equation $$x^3 - x = k - k^3 .$$ Obviously, $x=-k$ is a solution. Is it the only one? How to prove it?
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0answers
37 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
59 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
vote
0answers
97 views

A WolframAlpha error?

Consider the equation: $$ \dfrac{1}{\sqrt[3]{(x+3)^2}}-\dfrac{1}{\sqrt[3]{x^2}}=0 $$ that has the solution $x=\dfrac{-3}{2}$ as can be easely verified. But WolframAlpha gives no solutions (here). ...
1
vote
2answers
93 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
80 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
84 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 ...