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
86 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
128 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
5k views

How to calculate the percentage of increase/decrease with negative numbers?

I feel like an idiot for asking this but i can't get my formula to work with negative numbers assume you want to know the percentage of an increase/decrease between numbers ...
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2answers
154 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
353 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
125 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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1answer
118 views

Is there a general product formula for $\sum\limits_{k=1}^{n} k^p$

I'm familiar with Faulhaber's formula to express this sum as a much simpler one, but it appears that for any $p$ there's a product formula in $n$ for the sum e.g.: $$\begin{align} & ...
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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
113 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
393 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
187 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
175 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
48 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
83 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
550 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
139 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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268 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
284 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 ...
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4answers
118 views

Prove the inequality $|xy|\leq\frac{1}{2}(x^2+y^2)$

How can I prove the inequality $|xy|\leq\frac{1}{2}(x^2+y^2)$ I have tried substitute $x,y$ for numbers, which turns out right, but I don't know how to reason here. Thanks in advance!
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4answers
474 views

Limit of a Recursive Sequence

I'm having a really hard time finding the limit of a recursive sequence - $$ \begin{align*} &a(1)=2,\\ &a(2)=5,\\ &a(n+2)=\frac12 \cdot \big(a(n)+a(n+1)\big). \end{align*}$$ I proved ...
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2answers
67 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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2answers
336 views

$\frac1a+\frac1b+\frac1c=0 \implies a^2+b^2+c^2=(a+b+c)^2$? [closed]

How to prove that $a^2+b^2+c^2=(a+b+c)^2$ given that $\frac1a+\frac1b+\frac1c=0$?
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2answers
60 views

Problem Solving using Algebra

If Peter is $7$ years older than Sharon and John is twice as old as Peter, work out how old Peter is if the average of their ages is $19$. Thanks! :)
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1answer
145 views

Raising each side of a limit to a power?

If you have $\lim_{x\to \infty} f(x)=L$ $\quad$(or as $x \rightarrow c$ for that matter), is it correct to say that $\lim_{x\to \infty} f(x)^{g(x)}=L^{\lim_{x\to \infty}g(x)}$ as long as $L \not= 1$? ...
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1answer
123 views

I just found out that $0^0$ equals $1$, why is this? [duplicate]

I have done a lot of math so far, but I never stumbled on something this simple and yet mind boggling. Can someone tell me why $0^0$ equals $1$? I always knew that everything raised to a power of $0$ ...
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5answers
183 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. ...
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2answers
517 views

Distance between point and a line - problems simplifying the minimised distance equation

Someone asked how to prove the distance between a point $(x_1,y_1)$ and a line $Ax + By + C = 0$ is:$$\text{Distance} = \frac{\left | Ax_{1} + By_{1} + C\right |}{\sqrt{A^2 + B^2} }$$ The currently ...
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1answer
46 views

How to determine a function that satisfies the following:

This question is to determine a function which will later help me answer this question: Given this set of points, how do I determine an equation which satisfies them: n ...
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1answer
183 views

A finite sum of trigonometric functions

By taking real and imaginary parts in a suitable exponential equation, prove that $$\begin{align*} \frac1n\sum_{j=0}^{n-1}\cos\left(\frac{2\pi jk}{n}\right)&=\begin{cases} 1&\text{if } k ...
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2answers
171 views

Show $ \frac{x}{xy+x+1}+\frac{y}{yz+y+1}+\frac{z}{zx+z+1}=1 $ given $xyz = 1$

Please help me prove the equality: If $xyz=1$, prove that $$ \frac{x}{xy+x+1}+\frac{y}{yz+y+1}+\frac{z}{zx+z+1}=1 $$
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2answers
62 views

How to find $a,b\in\mathbb{N}$ such that $c = \frac{(a+b)(a+b+1)}{2} + b$ for a given $c\in\mathbb{N}$

Suppsoe that $$c = \frac{(a+b)(a+b+1)}{2} + b$$ Now $c$ is given - how does one find satisfying $a, b$?
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3answers
283 views

Finding the $x$ and $y$ values such that the partial derivatives are zero simultaneously

$f(x,y) = x^2 + 4xy + y^2 -4x + 16y + 3$ So, I proceeded with taking the partial derivatives: $f(x,y)_x = 2x + 4y - 4$ and $f(x,y)_y = 4x + 2y + 16$ and $f(x,y)_x = f(x,y)_y = 0$ $2x + 4y - 4 = ...
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2answers
45 views

Creating an application with a level system and I would like to convert the values into an equation

Here is what I have so far: you start at level $0$ with $0$ XP. The objective is to gain XP to level up. Once you reach $100$ XP you get to level $1$, $300$ XP = Level $2$, $600$ XP = Level $3$, ...
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1answer
135 views

Another trigonometric equation

Show that : $$31+8\sqrt{15}=16(1+\cos 6^{\circ})(1+\cos 42^{\circ})(1+\cos 66^{\circ})(1-\cos 78^{\circ})$$
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3answers
229 views

Area Between Curves

The problem I am working on is, "In Exercises 17 and 18, find the area of the region by integrating (a) with respect to and x (b) with respect to y." The two functions: $g(y)=4-y^2$, and $f(y)=y-2$ ...
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1answer
55 views

Find expotential function from two points

Clever people on this place, I'm having trouble with this, and I'm not able to see why what I'm doing is wrong... Here are two points: $(3,1)$, $(-1,16)$ And this is what my calculations are: ...
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3answers
126 views

How do we solve the system of equations?

How do we solve the system of equations? \begin{equation*} \begin{cases} 47a^3+129ab^2-93ba^2-83b^3-154a^2-178b^2+276ab = 0,\\ 2a^2+5b^2=23. \end{cases} \end{equation*}
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1answer
628 views

Proof of Remainder Theorem for polynomials

Lemma: Prove that for any polynomial function $f$ and any number $a$, there is a polynomial function $g$ and number $b$ such that $f(x) = (x-a)g(x) + b$ for all $x$. Proof: (Prove by strong ...
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1answer
170 views

translate and rotate coordinates

I have two dataset with Cartesian data points. I would like to uniformly translate the coordinates of one dataset to be properly rotated and on top of the other (having a certain max and min for all ...
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3answers
161 views

Equations - Solving for x

I have this problem: $$9x^3 - 18x^2 - 4x + 8 = 0$$ However, I'm not sure how to find the values of $x$. I moved the 8 over and factor out an $x$, but the trinomial it created can't be factored. ...
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2answers
484 views

Cubic polynomial discriminant

I am stuck on a cryptography problem that pertains to Elliptic Curves. The problem is stated as follows: Assume the cubic polynomial $X^3+AX+B = (X-a)(X-b)(X-c)$ If $4A^3 + 27B^2 = 0$, then show ...