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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2answers
85 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 ...
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1answer
101 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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2answers
48 views

Changing the cartesian coordinate system

The cartesian coordinate in 3D is given as: Are we allowed to make our own coordinate system (switching axes around). The question is can we change the axes around? ** Like: **LOOKING AT THE ...
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0answers
58 views

Find n that satisfies the following [duplicate]

Find the smallest positive integer n that satisfies the following: We can color each positive integer with one of n colors such that the equation w + 6x = 2y + 3z has no solutions in positive integers ...
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1answer
55 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 ...
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4answers
544 views

Show that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq \frac{9}{a+b+c}$ for positive $a,b,c$ [duplicate]

Show that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq \frac{9}{a+b+c}$, if $a,b,c$ are positive. Well, I got that $bc(a+b+c)+ac(a+b+c)+ab(a+b+c)\geq9abc$.
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1answer
41 views

Simplify this fraction with square roots; application to arctangent equation.

I need your help. I don't know how to simplify: $\frac{-1+\sqrt{3}+\sqrt{4+2\sqrt{3}}}{2\sqrt{3}} $ and $\frac{-1+\sqrt{3}-\sqrt{4+2\sqrt{3}}}{2\sqrt{3}}$ Thank you in advance. I found $1$ and ...
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3answers
65 views

How do I factor $\ t^4-2 \ $?

This binomial is part of a bigger problem that I need to solve, however, I am little stuck on how to factor it. $(t-1)(t-1)(t+1)$ does not work.
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2answers
37 views

Prove the given condition from given two quadratic equation

Question: If the quadratic equations $x^2+bx+c=0$ and $bx^2+cx+1=0$ have a common root then prove that either $b + c + 1 = 0$ or $b^2 + c^2 + 1 =bc + b + c$ Till yet, I had figured the common ...
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5answers
98 views

Proving $9$ divides $n^3 + (n+1)^3 + (n+2)^3$ [duplicate]

I'm trying to prove by MI. I have already distributed n+1, but now I'm stuck on how I can show 9 divides the RHS since $42n$ and $3n^3$ does not divide evenly. ...
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3answers
113 views

Problem 17 in chapter 3 of Spivak book

If $f(x)=0$ for all $x$, then f satisfies $f(x+y)=f(x)+f(y)$ for all $x$ and $y$, and also $f(xy)=f(x)f(y)$ for all $x$ and $y$. Now suppose that $f$ satisfies these two properties, but that $f(x)$ is ...
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2answers
315 views

How to find the equation of the graph reflected about a line?

Consider the graph of $y = e^x$ (a) Find the equation of the graph that results from reflecting about the line $y = 4$. (b) Find the equation of the graph that results from reflecting ...
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8answers
149 views

Inequality: $x^2+y^2+xy\ge 0$

I want to prove that $x^2+y^2+xy\ge 0$ for all $x,y\in \mathbb{R}$. My "proof": Suppose wlog that $x\ge y$, so $x^2\cdot x\ge x^2\cdot y\ge y^2\cdot y=y^3$ (because $x^2\ge 0$ so we can multiply ...
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2answers
65 views

Inequality on product of two positive numbers

This question is linked to this other question about a proof of the AM-GM inequality. All that I still don't understand, because I don't know how to prove, is the Lemma, by which if $a, b, c \in ...
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0answers
62 views

Proving uniqueness

Sorry this is such a trivial question, but I'd like to check my understanding on this subject before proceeding. Suppose that we have the quantity $x+a=b$ where $x$ is a variable, and we wish to show ...
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4answers
102 views

How many different positive integer factors does have?

How many different positive integer factors does $(2^7)(3^4)(7^3)(23^5)$ have? Do we have to do any combinations between the powers here?
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4answers
1k views

How to solve equations involving modulus function of the type $|x+1| - |1-x|=2 $ and $ |x-1|=|x|+a$?

I am able to solve equation of the type $ |5x+1|=|11-2x|$. I square both the side and my equation becomes $ (5x+1)^2=(11-2x)^2 $ further simplification gives me $ (5x+1)=\pm (11-2x)$. I get have ...
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2answers
160 views

Find parameters of short geometric series

I occasionally host D&D~ish game and having a way to do this will definitely improve my ability to serve better games, so when they want to upgrade their fireball 5 times, I can do it without the ...
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1answer
88 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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3answers
95 views

Partial fraction decomposition of a complicated rational function

Find the partial fraction decomposition of the rational function $\displaystyle \frac{2x^3+7x+5}{(x^2+x+2)(x^2+1)}$ I have tried dividing first but keep running into problem after problem, please ...
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1answer
28 views

Non-homogenous System where did I go wrong?

Solve the system $\vec{x^{'}}=\begin{pmatrix}2 & -5\\1 & -2 \end{pmatrix}\vec{x}+ \begin{pmatrix} -\cos t\\ \sin t \end{pmatrix}$ The Eigenvalues are $(2-\lambda)(-2-\lambda)+5=0 \implies ...
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3answers
214 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 $$ ...
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2answers
152 views

How to solve $(x-2)^4+(x-3)^4=1$?

I'm trying my hand on these types of expressions. how to solve $x$ and $y$ in $\displaystyle(x-2)^4+(x-3)^4=1$ and $\displaystyle (y-1)(y-2)(y-3)(y-4)=1$? please write step by step solution ! ...
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2answers
72 views

what problem do quadratics solve?

Areas of Math were developed to deal with problems which couldn't be tackled with preexisting methods. So, for example, 2D geometry was used to solve measuring problems with land, basic algebra for ...
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4answers
120 views

Series question

I was trying this question: Consider the infinite series $$\frac{1}{1!} +\frac{4}{2!}+\frac{7}{3!}+\frac{10}{4!}+...$$ If the series continues with the same pattern, find the an expression for the ...
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2answers
90 views

Describing asymptotic behaviour of a function

For question B! x^2+x+1/x^2 = 1+ [x+1/x^2] shouldnt the answer be asymptote at x=0 and y=1 ?? i dont understand the textbook solution
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3answers
130 views

how do you factor $x^2 +kx+40$ over the integer

please please help me, I'm having a lot of troubles. I tried to use a^2+2ab+b^2 formula (like i was told) but that's where get lost. I understand that Factoring uses the opposite operation, but 40 ...
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2answers
155 views

How to prove that certain integers and xy are solutions for a relation?

I am trying to solve the following problem: Let A be the set of all integers of the form a^2 + b^2 + 4ab where a and b are integers. Prove: a. if x and y are in A, prove xy is in A. b. Prove 121 is ...
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1answer
88 views

Solve for $a$ in the formula $\frac{x^a}{a}=b$

Rearrange this formula to find $a$ in terms of $x$ and $b$: $$\frac{x^a}{a}=b$$ So far I can rearrange up to here, but I don't know what to do next to get $a$ by itself: $$x^a=ab$$ ...
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0answers
76 views

Solve for x in $\tan x=2x$ [duplicate]

Is there a way to solve for $x$ for $\tan x=2x$? My cousin asked me about this and I wondering if there was some sort of trig identity I had to know.
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1answer
367 views

$\sin^2 \alpha + \sin^2 \beta - \cos \gamma < M$ given that the sum of the angles is $\pi$

Question: Find the least real value of $M$ such that the following inequality holds: $$\sin^2 \alpha + \sin^2 \beta - \cos \gamma < M$$ Given that $\alpha, \beta, \gamma \in \mathbb{R}^+$, ...
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2answers
126 views

Periodic sequence [duplicate]

$(x_n)_n$ is a sequence defined by the relation: $x_{n+2}=|x_{n+1}-x_{n-1}|$ for $n\geq1$ and $x_0,x_1,x_2$ are non-negative integers, not all three equal 0. I think this sequence is periodic, so here ...
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2answers
77 views

Solve for $X$: $4+2X > 7$

Solve for $$4+2x>7$$. My answer was 1 and 1/2 but I am not so sure it is right, could anyone confirm it for me. i need someone to solve the inequality. the first step would be to subtract $4$ ...
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0answers
114 views

Trivial question about a double summation

Let's say I have a function $f$ defined in $\{1,2,...,n\}\times \{1,2,...,m\}$ by $f(i,j)=ij$, is it immediately obvious that \begin{equation} \sum_{i=1}^n\sum_{j=1}^m f(i,j)=\sum_{i=1}^n\sum_{j=1}^m ...
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3answers
398 views

Find all the real Zeros of the function?

Please help me I'm stuck and I don't know how to go about this. :(
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2answers
76 views

Left/Right inverses of functions.

I am currently studying functions in general, and I've come across left and right inverses, however I can't wrap my head around the following: $f(x) = 3x^4$ I know this function doesn't have an ...
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2answers
192 views

How to find the zeros of an equation of nth degree

I was working on problems in my math textbook and I saw this problem as a side note and I couldn't figure it out. The author states: ...
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4answers
82 views

Solving an irrational equation

Solve for $x$ in: $$\frac{\sqrt{3+x}+\sqrt{3-x}}{\sqrt{3+x}-\sqrt{3-x}}=\sqrt{5}$$ I used the property of proportions ($a=\sqrt{3+x}$, $b=\sqrt{3-x})$: ...
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3answers
179 views

Need an algebra book

Since my last question was too general,I decided to delete it and write this one. The topics it must constain: linear equations linear inequalities graphing and analyzing linear functions systems ...
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1answer
49 views

Solving unknown powers for a super quadratic ellipse

$$\left|\dfrac xa\right|^m + \left|\dfrac yb\right|^n = 1$$ if $x$, $a$, $y$, $b$ are known, how do you solve for $m$ and $n$? Thank you.
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0answers
98 views

Richardson's theorem for constants

It's known that there is no algorithm for deciding for any elementary function is it identically zero or not (http://en.wikipedia.org/wiki/Richardson%27s_theorem ). But if I consider only constants - ...
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4answers
84 views

$x^2+y^2=1, 5x+12y+13=0$ Simultaneous Equations

Can someone solve this for me and show working out? I just can't do it and I don't know why I am getting x and y wrong. It will be very much appreciated. As basic as possible as well please.
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2answers
600 views

Help find sum to infinity of a series - odd numbers with a common ratio

I am trying to derive the formula for the variance of a geometric distribution and am stuck at the following problem: I need to find the sum to infinity for the following series: ...
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3answers
52 views

Algebraic simplification

I have never learned this in school, I only learned algebra when you have $x$ and numbers, in equations, like this: $$2x = 5(-2 + 5x)^2$$ I can solve that, but I cannot solve this one: $$-3(7 ...
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1answer
1k 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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3answers
143 views

How can I use the distributive property to rewrite an algebraic fraction?

I have an expression: $$N\left(\dfrac{N(N+1)(N-1)+3N}{3}\right)$$ Can can I use the distrubtive property to form: $$N^2\left(\dfrac{(N+1)(N-1)+3}{3}\right)$$ If so, how? Could someone advise me on ...
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3answers
74 views

Value and simplify

I want to find the value and simplify square root 36 ? Square root of 36 is 6 But I would know how to find the value and simplify it .
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2answers
269 views

How to get all solutions to equations with square roots

I would like to find all solutions to $$b-a\sqrt{1+a^2+b^2}=a^2(ab-\sqrt{1+a^2+b^2})$$ $$a-b\sqrt{1+a^2+b^2}=b^2(ab-\sqrt{1+a^2+b^2})$$ I found some solutions. For example, $a = 1, b = \pm i$ and ...
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2answers
1k views

Domain and Range of f(x)

Find the domain and range of Function? $$F(x) = \frac{1}{\sqrt{25-x^2}}$$ Domanin f(x) has real value , if $$25-x^2\geqslant0$$ $$-x^2\geqslant-25$$ $$x^2\le25$$ $$-5\le x\le5$$ $$D_f= [-5,5]$$ let ...
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3answers
305 views

Deriving an equation of a parabola

I would like to understand the concept of deriving an equation, given values. E.g. Derive the equation of parabola whose vertex is at origin and focus $(-3,0)$. From this, I reckon the ends are ...