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.

learn more… | top users | synonyms (2)

3
votes
0answers
39 views

Prove that $s(n-1)s(n)s(n+1)$ is always an even number

Let $n$ be a natural number, and let $s(n)$ denote the sum of all positive divisors of $n$. Show that for any $n>1$ the product $s(n-1)s(n)s(n+1)$ is always an even number. I calculated the sum of ...
0
votes
1answer
31 views

ASTC: Finding exact values of trigonometric functions

Our teacher showed us this really dodgy way of finding exact values by drawing up the 4 ASTC (all stations to central diagram) quadrants and making a right angle to the x axis. So how would I do a ...
1
vote
0answers
28 views

About a matrix identity.

In a document named as "The Matrix Cook-Book" I saw two expressions of which I do not get any clue how they are derived. For $n = 3:$ $\det(I + A) = 1 + \det(A) + Tr(A) + 1/2\ Tr(A)^2 − 1/2\ ...
-2
votes
3answers
32 views

write an expression [on hold]

A word processor determines the width of the body of text on a page. The page is 11 inches wide and has two equal size margins of x inches on each side of the text. Write a formula that gives the ...
3
votes
3answers
77 views

Why does $e^{-x}$ approach $0$ as $x$ gets large? [on hold]

Why is it that $$\lim_{x \to -\infty} e^x = 0?$$
2
votes
1answer
218 views

Probability or Set

I'm really good at probability, but this time I seems like I'm not. My friends asked me a very tricky question, and I want to see if there's anyone who can find out the answer. Here's the ...
2
votes
6answers
55 views

Limit of $\lim_{x \rightarrow 0} \frac{\sin xy^2}{x}$

Limit of $$\lim_{x \rightarrow 0} \frac{\sin xy^2}{x}$$ I know (thanks to wolfram) it is equal to $y^2$, but i do not know how to show that.
1
vote
2answers
47 views

trying to solve $\sqrt{\cos(x)-2\cos(2x)}+\sqrt{2}\cos(2x)=0$

The equation is $$\sqrt{\cos(x)-2\cos(2x)}+\sqrt{2}\cos(2x)=0$$ The system is $$ \begin{cases} \cos(x)-2\cos(2x)=2\cos^2(2x) \\ -\sqrt{2}\cos(2x)\ge 0 \iff \cos(2x)\le 0 \end{cases} $$ The ...
0
votes
2answers
35 views

Find the limit of $\frac{x+y+\sin xy}{x^2+y^2+\sin^2 (xy)}$

Find the limit of: $$\lim_{(x,y)\rightarrow(+\infty, +\infty)}\frac{x+y+\sin xy}{x^2+y^2+\sin^2 (xy)}$$ I think the solution could be: $$\frac{x+y+\sin xy}{x^2+y^2+\sin^2 (xy)} \le \frac{x+y+\sin ...
1
vote
1answer
45 views

Algebra Problem: Division

Can someone help me with a problem involving the expression $$\frac{(2x^3-3x^2+b)}{(4-x^2)}?$$ The question is to find which values $b$ can be to simplify the expression, but I do not know how to ...
0
votes
8answers
90 views

$x(x^2-2)=0$, The answers are $x = 0, \sqrt{2}$, how do I get there? [on hold]

$$x(x^2-2)=0$$ The answers are $x=\sqrt{2}, 0$ how do I get there?
3
votes
2answers
24 views

Minimum of $f(x)=\sum_{i=1}^n\frac{a_n}{x-b_n}$ occurs at extreme point?

Let $a_1,\ldots,a_n$ be real numbers and $b_1,\ldots,b_n>1$. Define $$f(x)=\sum_{i=1}^n\frac{a_i}{x-b_i}.$$ Is it always true that $f(x)\geq\min\{f(0),f(1)\}$ for all $x\in[0,1]$?
0
votes
2answers
26 views

Changing equation to x equals

Im currently stuck on this equation I need to modify to be in terms of x $$y=-x^2+4$$ I got something like this which looks wrong $$x = -\sqrt{y+4}$$ First you would subtract the 4 from both ...
1
vote
3answers
33 views

Distance/Speed word problem

A train of length 300m can cross a pole in 8 seconds. How long will it take to cross a platform of length 600m. I can't seem to appreciate the very beginning. Crossing the pole implies that the time ...
1
vote
2answers
26 views

An 11-gon with complex numbers

Let $A_1 A_2 \dotsb A_{11}$ be a regular $11$-gon inscribed in a circle of radius $2$. Let $P$ be a point, such that the distance from $P$ to the center of the circle is $3$. Find $[PA_1^2 + PA_2^2 ...
0
votes
2answers
39 views

Geometric progression (compound interest)

"A man, who started work in 1990, planned an investment for his retirement in 2030 in the following way. On the first day of each year, from 1990 to 2029 inclusive, he is to place £100 in an ...
1
vote
2answers
80 views

Trying to solve $\sqrt{7-4\sqrt2 \sin x}=2\cos(x)-\sqrt2 \tan(x)$

The equation is $$\sqrt{7-4\sqrt2 \sin x}=2\cos(x)-\sqrt2 \tan(x)$$ We get the system $$ \begin{cases} 7-4\sqrt 2 \sin(x)=4\cos^2(x)-2\sqrt2\cos(x)\tan(x)+2\tan^2(x) \\ 2\cos(x)-\sqrt2 \tan(x)\ge 0 ...
-1
votes
1answer
29 views

Months for the amount paid to be equal? [on hold]

If I take my pension now I can get $\$2,000$ a month. If I wait $12$ months I can get $\$2,500$. How many months will it take for the total amount paid to me be equal?
0
votes
2answers
35 views

Help with a progress bar algorithm for a website

I have a progress bar in a website that needs to be filled based on the number $50$. So at $50$ it will be $100\%$ full. The problem is that it starts at about $20\%$ then follows this pattern : ...
7
votes
1answer
89 views

Find all pair of cubic equations

Find all pair of cubic equations $x^3+ax^2+bx+c=0$ and $x^3+bx^2+ax+c=0$, where $a,b$ are positive integers and $c$ not equal to $0$ is an integer, such that both the equations have three integer ...
4
votes
5answers
80 views

Does the limit $\lim\limits_{x\to0}\left(\frac{1}{x\tan^{-1}x}-\frac{1}{x^2}\right)$ exist?

Does the limit: $$\lim\limits_{x\to0}\frac{1}{x\tan^{-1}x}-\frac{1}{x^2}$$ exist?
1
vote
1answer
31 views

Solving two equations with 2 variables

I am wondering if this equations can be solved by "a" and "b": b = 1 + 0.31*a a = c1 - c2/b c1 and c2 are constants, but change depending on some initial assumptions. One example of their ...
0
votes
3answers
48 views

Simplifying Cube Roots Containing a Square Root

I was doing a problem today, and arrived at the (correct) answer of $x^3 = 16000\sqrt2$ Obviously I want to simplify this further. My text book jumps straight to $x = 20\sqrt2$ with no explanation. ...
2
votes
3answers
59 views

Give the equations that are a tangent to the parabola $y = x^2 + 5x + 6$ and pass through $(1,1)$

I have been given the question: Give the equations that are a tangent to the parabola: $y = x^2 + 5x + 6$ and pass through the point $(1,1)$ I have tried two different methods for solving this. ...
2
votes
4answers
71 views

Is there an integer solution to $x^2+1978=y^2$

Is there an integer solution to $x^2+1978=y^2$? Don't know really how to approach this. Thanks
0
votes
5answers
55 views

Are these two expression equal?

My friend insisted that $(-1)^{(-n)}$ is equivalent to $(-1)^n$ for any number of $n$. A quick check in the Wolfram Alpha show ...
3
votes
4answers
60 views

Trying to solve the trig equation $\sqrt{3+4\cos^2(x)}=\frac{\sin(x)}{\sqrt 3}+3\cos(x)$

The equation is $$\sqrt{3+4\cos^2(x)}=\frac{\sin(x)}{\sqrt 3}+3\cos(x)$$ My solution goes like this $$ \begin{cases} 3+4\cos^2(x)=\frac{\sin^2(x)}{3}+\frac{6}{\sqrt 3}\sin(x)\cos(x)+9\cos^2(x) \\ ...
0
votes
2answers
31 views

Getting two different sets of results for $\sqrt{17+7\sin(2x)}=3\sin(x)+5\cos(x)$

The equation is $$\sqrt{17+7\sin(2x)}=3\sin(x)+5\cos(x)$$ My solution is, first, to define a system: $$ \begin{cases} 17+7\sin(2x)=(3\sin(x)+5\cos(x))^2 \\ 3\sin(x)+5\cos(x)\ge 0 \end{cases} $$ ...
0
votes
2answers
32 views

What is wrong with this formula?

I'm trying to make a formula that converts an ellipse in general form to one in standard. My steps to derive it are as follows: $$ax^2+bx+cy^2+dx+e=0$$ Move e to the other side... ...
2
votes
0answers
33 views

Let $A_1 A_2 \dotsb A_{11}$ be a regular 11-gon inscribed in a circle of radius 2.

Let $A_1 A_2 \dotsb A_{11}$ be a regular 11-gon inscribed in a circle of radius 2. Let $P$ be a point, such that the distance from $P$ to the center of the circle is 3. Find $PA_1^2 + PA_2^2 + \dots ...
-1
votes
0answers
37 views

Equilateral triangle [on hold]

An equilateral triangle is one in which all three sides are of equal length. If two vertices of an equilateral triangle are $(0,\,4)$ and $(0,\,0)$, find the third vertex. How many triangles are ...
0
votes
0answers
40 views

Solving three quadratic simultaneous equations with three variables

I need to solve the following simultaneous equations: $$(2-a)^2+(3-b)^2+(-5-c)^2=6$$ $$(1-a)^2+(2-b)^2+(-3-c)^2=6$$ $$a+b+c=0$$ I've tried expanding and doing it the long way, but I don't ...
2
votes
3answers
91 views

Computing $\sqrt[3]{1\,}$

I know that the answer is always $1$, but they are looking for some way to get to that answer and I don't know what it is. I am not good at english math terms, but maybe it has to do with differential ...
2
votes
1answer
54 views

How to show $\binom{2n}{n} \ge \prod_{n < p \le 2n} p $?

What is the best way to show \begin{equation} \binom{2n}{n} \ge \prod_{n < p \le 2n} p \end{equation} for prime $p$. I know that $ 2^{2n} = (1+1)^{2n} \ge \binom{2n}{n}$. and \begin{equation} ...
2
votes
1answer
62 views

Difficult sets of Equations, counting

Let $ m$ be the number of solutions in positive integers to the equation $ 4x+3y+2z=2009$, and let $ n$ be the number of solutions in positive integers to the equation $ 4x+3y+2z=2000$. Find the ...
3
votes
1answer
46 views

Find the sum of the roots of the floor equation

How to find the sum of the roots of the following floor equation? $$[\frac{x}{2}]+[\frac{x}{3}]+[\frac{x}{5}]=x$$ I found the following solutions by Mathematica: $\{\{ x= 0\},\{x = 6\},\{x = ...
0
votes
0answers
48 views

Find the value of $\frac {a+b+c}{x+y+z}$

$a^2+b^2+c^2=15\space \space$ $x^2+y^2+z^2=25$ $ax+by+cz=10$ Find the value of $\frac {a+b+c}{x+y+z}$ Thanks for any help.
0
votes
0answers
55 views

why $\frac{a}{b}\pmod p=\frac{a\pmod p}{b\pmod p}$

It is said this following is theorem? what's this name? and How to prove it? Thanks show that $$\dfrac{a}{b}\pmod p=\dfrac{a\pmod p}{b\pmod p},a,b\in N^{+},(a,p)=1,(b,p)=1$$
0
votes
2answers
42 views

Linear Equation in 4 variables- No of solutions

If 3a+6b+9c+4d = 100 and a ,b,c and d are natural numbers , then how many values d can take? How to approach this type of problem?
0
votes
0answers
35 views

What method(s) can be employed to solve this equation?

Solve for $n$ in the following equation: $$ 0 = 1780*1.006^n - 37n $$ Here's what I've tried: $$\begin{align*} 0 &= 1780*1.006^n - 37n \\ 37n &= 1780*1.006^n \\ \frac{37n}{1780} &= 1.006^n ...
15
votes
3answers
2k views

How to solve equations to the fourth power?

Is it possible to manually retrieve the value of $y$ from the following equation $$153y^2-y^4=1296$$ WolframAlpha has four solutions for $y$: $-12, -3, 3, 12$. How has it solved? What I've achieved ...
1
vote
4answers
63 views

Trying to solve $\sqrt{2\cos^2(x)-\sqrt{3}}+\sqrt2 \sin(x)=0$

The equation is $$\sqrt{2\cos^2(x)-\sqrt{3}}+\sqrt2 \sin(x)=0$$ I solve it thus: $$ \begin{cases} 2\cos^2(x)-\sqrt3=2\sin^2(x) \\ -\sqrt2 \sin(x)\ge 0 \iff \sin(x)\le 0 \end{cases} $$ The first ...
-5
votes
0answers
16 views

prove the given question [on hold]

Prove that $\sec(2 \alpha)\cos(45^{\circ}-\alpha)\sin(45^{\circ}+\alpha) = \dfrac{1}{2}$.
-2
votes
1answer
32 views

Choose a variable to represent the number in parentheses.. [on hold]

The distance traveled in 3 h of driving was 210 km. ( hourly rate).... also write an equation that represents the given information
-3
votes
0answers
36 views

Speed/Distance math problem [on hold]

A corrections Canada transportation vehicle needs to travel $660$km. The vehicle travels at $100$km an hour for two hours. The driver stopped for gas for approximately $30$ minutes and when resumed ...
1
vote
0answers
16 views

Writing a word problem as a function.

I would like to verify that this word problem was translated into a function correctly. A towing company charges a flat rate of $100.97$ dollars per day plus $0.81$ dollars per mile. The ...
0
votes
1answer
32 views
+50

Comparing Coefficients

If I have the equation: $4m(m-1)x^m .\sum_{i\geq 0}a_ix^i+x^m.\sum_{i\geq 0}a_ix^i=0$ ; $a_0\neq 0$ why am I able to say that $4m(m-1)+1=0$? I would understand if the equation rather than being an ...
1
vote
6answers
122 views

Prove $((a+b)/2)^n\leq (a^n+b^n)/2$

Struggling with this proof. Prove that $$\left(\frac{a+b}{2}\right)^n≤\frac{a^n+b^n}{2},$$ where $a$ and $b$ are real numbers such that $a+b≥0$ and $n$ is a positive integer. What technique would ...
3
votes
2answers
47 views

Polynomial with real roots

Consider the polynomial: $$f=X^4+4X^3+6X^2+aX+b$$ We know that $f$ has four real roots. Let $x_1,x_2,x_3,x_4$ be the roots of this polynomial. How can one compute ...
0
votes
2answers
76 views

How many pairs $(m, n)$ exist?

For certain pairs $ (m,n)$ of positive integers with $ m\ge n$ there are exactly $ 50$ distinct positive integers $ k$ such that $ |\log m - \log k| < \log n$. Find the sum of all possible ...