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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1answer
131 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
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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
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4answers
312 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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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
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1answer
532 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
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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
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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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2answers
664 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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1answer
185 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
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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
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3answers
891 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
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: ...
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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?
2
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3answers
359 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
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
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1answer
946 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
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2answers
415 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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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!
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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 ...
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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
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
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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 $\mathbb{Z}_5$. ...
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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). ...
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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 ...
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1answer
78 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 ...
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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 ...
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3answers
79 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 ...
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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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3answers
51 views

Best argument to prove $|x|\le a \iff -a\le x \le a$

$$|x|\le a \iff -a\le x \le a$$ I can only verify the integrity of this by talking about distances on the number line. But is there a algebraic argument that proves this?
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0answers
319 views

Calculating the interest rate for an annuity (Exam FM)

I have been searching for a way to solve for the interest rate given the monthly payments of a loan. I would like to set up a problem as the following. $X$=monthly payment , $i$=effective ...
1
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2answers
512 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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3answers
1k views

surjective, but not injective linear transformation

$T$ is a transformation from the set of polynomials on $t$ to the set of polynomials on $t$. So, the input to $T$ should be a polynomial, and the output should be some other polynomial. Two common ...
1
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0answers
69 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 ...
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6answers
412 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
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1answer
103 views

Composition of a piecewise and non-piecewise function

Say you have 2 functions, one of which being a piecewise function: $f(x)= x^2+2, x<1$ or $2x^2+2, x>=1$ And the other: $g(x)=x^4+1$ How would you find the $f[g(x))]$? I understand regular ...
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5answers
196 views

Solving $x^3 + x^2 - 4 = 0$

Does anyone know how to solve $$ x^3 + x^2 - 4 = 0 $$ analytically? That is, without using numerical methods to attain an approximate solution.
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1answer
2k views

Intersection of a plane with an infinite right circular cylinder by means of coordinates

So, I started studying analytic geometry and I must say I'm finding it much harder than "classic" geometry, because of the equations without help from diagrams... Still, I wanted to see how to use it ...
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1answer
89 views

integer ordered pair,s $(x,y)$ in $1!+2!+3!+…+x! =y^3$

(1) Total no. of integer ordered pair,s $(x,y)$ in $1!+2!+3!+............+x! =y^2$ (2) Total no. of integer ordered pair,s $(x,y)$ in $1!+2!+3!+............+x! =y^3$ (3) Total no. of integer ordered ...
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1answer
162 views

Please help to complete proof of inclusion and exclusion principle

I want to complete the following proof: So continuing where the author left off, I did the following: $\begin{array} {cc} & \sum\limits_{i=1}^{n-1} P(A_i) - \sum\limits_{1\le i < j \le ...
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1answer
572 views

using the same symbol for dependent variable and function?

Is it wrong to represent a dependent variable and a function using the same symbol? For example, can we write the parametric equations of a curve in xy-plane as $x=x(t)$, $y=y(t)$ where $t$ is the ...
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3answers
161 views

Evaluating $\sum_{i=0}^k \frac{(-1)^{k-i}k! (n+i)^{k-1}}{i!(k-i)!}$

How to evaluate the the following sum where $n $ is an integer greater than $0$. $$\sum_{i=0}^k \frac{(-1)^{k-i}k! (n+i)^{k-1}}{i!(k-i)!}$$ I think the answer is $0$, but I can not prove it.
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4answers
620 views

Parabola and Circle problem : The parabola $y =x^2-8x+15$ cuts the x axis at P and Q. A circle is drawn …

Problem : The parabola $y=x^2-8x+15$ cuts the x axis at P and Q. A circle is drawn through P and Q so that the origin is outside it. Find the length at a tangent to the circle from O. My approach ...
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2answers
72 views

Finding $x$ from inequality: $\left | \frac{3^n + 2}{3^n + 1} - 1 \right | \le \frac{1}{28}$

Find $x$ in $\mathbb{Z}$ satisfying this inequality: $$\left | \frac{3^n + 2}{3^n + 1} - 1 \right | \le \frac{1}{28}.$$ I tried something, but I don't think it's correct. $$-\frac{1}{28} ...
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1answer
84 views

How to find the sine of an angle

How to find the sine/cos/tangent/cotangent/cossec/sec of an angle: In degrees $\sin(23^{\circ}) =$ ? In radians $\sin(0.53) =$ ?
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5answers
197 views

Is $\frac 1 0$ undefined or equal to $\tilde{\infty}$? [duplicate]

Is $\displaystyle\frac 1 0$ undefined or equal to $\tilde{\infty}$? I know that $\displaystyle\lim_{x\to0}\frac 1 x=\tilde{\infty}$, how about $\displaystyle\frac 1 0$? Thank you. p.s. ...
1
vote
1answer
69 views

How to factor $2b^2c^2 + 2c^2a^2 + 2a^2b^2 -a^4-b^4-c^2$?

The term is: $2b^2c^2 + 2c^2a^2 + 2a^2b^2 -a^4-b^4-c^2$ And the answer is : $(a+b+c)(b+c-a)(c+a-b)(a+b-c)$ I have tried a lot, but could't accomplish. Please don't bring up any complex method, it ...
1
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3answers
188 views

Why does tan(t) touch the unit circle at (1,0)?

I can't get my head around this, any help would be very much appreciated. Thanks EDIT: t is an angle, where 0 < t < 90, angle t is in degrees EDIT: Added a picture I lifted from google