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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1
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3answers
25 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 ...
2
votes
0answers
11 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 $$f(x)=\sum_{i=1}^n\frac{a_n}{x-b_n}.$$ Is it always true that $f(x)\geq\min\{f(0),f(1)\}$ for all $x\in[0,1]$?
1
vote
2answers
62 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 ...
0
votes
2answers
21 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
39 views

How many of each ticket were sold in one day?

Child tickets - $\$7$ Adult Tickets - $\$10$ Senior Tickets - $\$5$ Day one sold $678$ tickets for $\$5,812$ Day two sold $535$ tickets for $\$4,541$ How many of each ticket were sold on day one ...
1
vote
1answer
34 views

Use of restriction to disallow aberrant series

My question concerns restrictions on the exercise of normal algebraic rules. The most well known restriction is the prohibition on division by zero (PDZ). This is justified by various 'proofs' of ...
1
vote
2answers
20 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
34 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 ...
0
votes
0answers
24 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 ...
7
votes
1answer
61 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 ...
0
votes
2answers
33 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 : ...
0
votes
2answers
293 views

Parametric equation of ellipse with foci at origin

I want to know what the parametric equation for an ellipse is if the one of the foci is centered at the origin. I know the semi-major and minor axes. I know the parametric equation of an ellipse ...
-1
votes
1answer
27 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?
4
votes
4answers
54 views
5
votes
2answers
96 views

When is $(a+b)^n \equiv a^n+b^n$?

I remember a relation like $(a+b)^n \equiv a^n+b^n$, but I don't remember mod which numbers this is true. Where can I learn more about this?
2
votes
4answers
63 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
3answers
43 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. ...
3
votes
1answer
62 views

Chance of Drawing All of a Subset

I have a simple question but I can't seem to find the answer anywhere. Say that I have a set $\mathbb Z$ and a subset of that $\mathbb X$. I want to draw elements from $\mathbb Z$ until there is at ...
2
votes
3answers
55 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. ...
1
vote
1answer
29 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 ...
1
vote
4answers
62 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 ...
0
votes
5answers
53 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 ...
0
votes
1answer
1k views

How to combine an amount of money with the compound interest function?

Tommy has some money at home from his graduation modeled by the function $h(x)=350$. He read about a bank that has savings accounts that accrue interest according to the function $s(x)= 1.04 ...
3
votes
4answers
59 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
29 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: $$ \begin{cases} 17+7\sin(2x)=(3\sin(x)+5\cos(x))^2 \\ 3\sin(x)+5\cos(x)\ge 0 \end{cases} $$ ...
1
vote
3answers
65 views

Find $LK_1^2 + LK_2^2 + \dots + LK_{11}^2$.

$K_1 K_2 \dotsb K_{11}$ is a regular $11$-gon inscribed in a circle, which has a radius of $2$. Let $L$ be a point, where the distance from $L$ to the circle's center is $3$. Find $LK_1^2 + LK_2^2 + ...
0
votes
2answers
31 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 ...
40
votes
7answers
3k views

Why $\sqrt{-1 \times {-1}} \neq \sqrt{-1}^2$?

I know there must be something unmathematical in the following but I don't know where it is: \begin{align} \sqrt{-1} &= i \\ \\ \frac1{\sqrt{-1}} &= \frac1i \\ \\ \frac{\sqrt1}{\sqrt{-1}} ...
14
votes
4answers
1k views

Every year, there is a contest…

Every year, there is a contest to see who has the heaviest pumpkins for that year. Last year, a farmer brought 5 pumpkins to the contest. Instead of weighing them one at a time, he informed the ...
-1
votes
0answers
36 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
57 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 ...
2
votes
1answer
53 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} ...
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 = ...
-4
votes
1answer
37 views

Rearrange and solve for $N: 16 = \frac{1}{n}\cdot 25 + \frac{n-1}{n} \cdot 218.75$

I need to solve for $N$ to get $16$ with the following formula, I'm very bad a re-arranging though, so does anyone have an answer to this? $$16 = \frac 1 n \cdot 25 + \frac{n-1} n \cdot218.75$$ ...
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 ...
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 ...
0
votes
0answers
47 views

Equation of a Line

I have following stuff $(x_1, y_1) = (3, 36)$ with $m = -10$, $c=66$ (got it from $x_2$, $y_2$) $(x_2, y_2) = (3.5, 31)$ with $m = -8$, $c=59$ (got it from $x_3$, $y_3$) Now I have another ...
-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
1
vote
5answers
114 views

Why is the graph of $x=y^2$ and $y=\sqrt{x}$ not the same?

So if you take $x=y^2$ and get the sqrt of both sides you get $y=\sqrt{x}$ so they are the same right? But when you graph them, $y=\sqrt{x}$ only shows the positive $y$ values because you can't sqrt a ...
-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 ...
3
votes
2answers
46 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 ...
1
vote
6answers
120 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 ...
1
vote
0answers
15 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 ...