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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3
votes
0answers
20 views

How does one solve $y^y-x^x=x$ for $x$ as a function of $y$?

In order to find the answer to this question I started thinking that as a first step to obtain the first and second column, one would have to solve the equation: $$y^y-x^x=x$$ for $x$ as a function ...
4
votes
1answer
29 views

$P(z)=0$ iff $Q(z)=0$, $P(z)=1$ iff $Q(z)=1$. Prove that $P(x)=Q(x)$ for all $x$

Assume $P(x)$ and $Q(x)$ are polynomials with complex coefficients with degree greater than or equal to $1$ such that $P(z)=0$ if and only if $Q(z)=0$, $P(z)=1$ if and only if $Q(z)=1$. Prove that ...
0
votes
2answers
17 views

Computing an academic grade when relative weights are changed

My grade is 88.6% (High B) and we get 80%(Assessment Grade) and 10%(Homework). My teacher is now making this 70%(Assessment Grade) and 30%(Homework). I have done all my homework 100% and I've been ...
2
votes
2answers
47 views

If 2 people pay 10 each, how much would a 3rd person have to pay to have an equal share?

If person 1 and 2 pay $\$10$ to equal $\$20$, how much would person 3 have to pay person 1 and 2 to become even? My solution: 20 divided by 3 is 6.66 so wouldn't the 3rd person just have to pay ...
2
votes
2answers
36 views

Find the number of children, given that the estate was divided evenly between them [on hold]

Problem of the Week at University of Waterloo: A man died leaving some money in his estate. All of this money was to be divided among his children in the following manner: $x$ to the first ...
-3
votes
3answers
33 views

The closed form sum of $12 \left[ 1^2 \cdot 2 + 2^2 \cdot 3 + \ldots + n^2 (n+1) \right]$… [on hold]

The closed form sum of $12 \left(1^2 \cdot 2 + 2^2 \cdot 3 + \ldots + n^2 (n+1) \right),n \geq 1$ is $n(n+1)(n+2)(an+b)$. Find $an + b$.
-6
votes
1answer
64 views

you know root square of -1, what is the larger of the square? [on hold]

there is a square ABDC, $BD = \sqrt{-1}$ what is the value of AB=BC=DC=AD?
1
vote
1answer
36 views

Find polynomial f(n) such that for all integers $n$ $\geq 1$, we have

Find polynomial f(n) such that for all integers $n \geq 1$, we have $3\left( 1\cdot2 + 2\cdot3 + \ldots + n(n+1) \right) = f(n)$. Write f(n) as a polynomial with terms in descending order of $n$.
0
votes
1answer
29 views

How to solve $D=\sqrt{X^2+MX^2}$ for $X$?

How I to solve $D=\sqrt{X^2+MX^2}$ for $X$? With my rudimentary experience, I find myself incapable. I apologize for asking a question after asking a similar one previously (several days ...
1
vote
2answers
67 views

Solve system of 3 equations

$x+y+z=0$ $x^2+y^2+z^2=6ab$ $x^3+y^3+z^3=3(a^3+b^3)$ this is what i reasoned out so far; $xyz=a^3+b^3$ $x^2+zx+z^2=3ab$ $y^2+zy+z^2=3ab$ $x^2+xy+y^2=3ab$ $y^2=3ab+zx$ $x^2=3ab+zy$ ...
1
vote
1answer
59 views

Four different positive integers a, b, c, and d are such that $a^2 + b^2 = c^2 + d^2$

Four different positive integers $a, b, c$, and $d$ are such that $a^2 + b^2 = c^2 + d^2$ What is the smallest possible value of $abcd$? I just need a few hints, nothing else. How should I begin? ...
0
votes
3answers
43 views

The meaning of dot centered vertically, as in $3\cdot 5$

I just went to a site to practise some maths as I am studying some maths on my OU course in computing and IT and I ran into some symbols that I did not understand The questions were solve ...
0
votes
1answer
16 views

Rearranging $ca^{b-1}/d^2$

I'm am try to rearrange $\frac{ca^{b-1}}{d^2}$ to $\large{\frac{c}{d^2a^{b-1}}}$ but I am having difficulty. I have tried times both top and bottom with various expressions such as $a^{b-1}$ but with ...
1
vote
0answers
30 views

Closed form for the summation $\sum_{k=1}^n\dfrac{1}{r^{k^2}}$

Is there any closed form for the finite sum $$\sum_{k=1}^n\dfrac{1}{r^{k^2}}$$ or infinite sum ( when it is convergent) $$\sum_{k=1}^\infty\dfrac{1}{r^{k^2}} ?$$ When I try to solve this problem, I ...
14
votes
5answers
1k views

Is there something special about 2015?

Is there some property which is satisfied only by the number 2015 (among natural numbers, say) or is there a relatively simple question for which the answer is, surprisingly, 2015? This is inspired ...
1
vote
1answer
15 views

Show that $dn^{\beta}/(\epsilon n^{1/2})^2$ can be written as $d /(\epsilon^2)(n^{(1-\beta)})$

Show that $dn^{\beta}/(\epsilon n^{1/2})^2$ can be written as $d /(\epsilon^2)(n^{(1-\beta)})$ I have tried $d n^{\beta}/(\epsilon^2) (n^{5/2})$ and then $dn^{(\beta-5/2)}/\epsilon^2$ But the 5/2 is ...
0
votes
2answers
26 views

The range of $\frac{2^x-1}{2^x+1}$

I am trying to find the range of the function $\frac{2^x-1}{2^x+1}$. If we draw using a graph plotter we can see that the range is $-2<y<2$. To find the upper bound, I tried ...
-2
votes
1answer
20 views

Can someone show me how this algebraic expression is worked out fully?

I'm not sure how they went from $\frac{k2i(1)2i(2)}{\frac{d}{8}}$ to 32F? I'm weak in algebra so if anyone has any reccomendations how I can improve in manipulating equations or websites and ...
0
votes
1answer
28 views

number of solutions of these equations.

Find the number of solution for this equation without drawing graph?! Total number of solutions for $2^{\cos x}=|\sin x|$ in $[-2\pi,5\pi]$ a) $14$ b) $15$ c) $16$ d) $17$ [ans given : ...
2
votes
1answer
24 views

Symmetric and homogeneous three variable inequality with radicals.

While trying to solve a problem, I got the following inequality which appears correct, but I cannot prove. For positive $x, y, z$, $$\sum_{cyc} \frac{x}{y^2+z^2} \ge \sum_{cyc} ...
-4
votes
1answer
37 views

How to solve: Differentiate d/dx 7^-1/2 [on hold]

All the question says Differentiate d/dx 7^-1/2
-4
votes
0answers
31 views

Trigonometric math problem [on hold]

A camera is mounted at a point 3000 ft from the base of a rocket launching pad. The Rocket rises vertically when launched, and the camera's elevation angle is continually adjusted to follow the bottom ...
1
vote
1answer
42 views

Roots less than 1 if at least one coefficient is greater than one

I have this doubt. If you have this equation with $\alpha_i \in \mathbb R$ $$P(z)=1-\alpha_{1}z-\alpha_{2}z^{2}- \cdots - \alpha_{p}z^{p}=0$$ I believe that if there exist an $\alpha$ greater or equal ...
0
votes
3answers
27 views

Beginner exponent/simplification question

Hey there I am having some trouble remembering all the old exponent rules and such, for example, $$ \frac{1}{(6+7^n) ^3} $$ How can I simplify this? I know that (7^n)^3 is the same as (7^3n), but ...
-1
votes
0answers
34 views

How does $\frac{x-(x+h)}{kx(x+h)}=\frac{-1}{x(x+k)}$

I'm sorry, I know this is very basic. But I'm getting lost somewhere in the algebra. :( Thank you
-1
votes
2answers
124 views

Is $x/x$ continuous at $0$? [on hold]

Just wondering, while studying limit, if $x\over x$ is continuous at $0$. $f(0)={0 \over 0}$ ,, but $x/x=1$. In this case, is it continuous at $0$?
1
vote
2answers
43 views

Combinatorics using a geometric diagram

How can I do this without trial-and-error? It has something to do with a triangle and summing the next row?
2
votes
1answer
41 views

Polynomial prove exercise

$P(x)=x^n + a_1x^{n-1} +\dots+a_{n-1}x + 1$ with non-negative coefficients has $n$ real roots. Prove that $P(2)\ge 3n$ I don't have an idea how to do that, I'm in 4th grade high school, you don't have ...
0
votes
2answers
32 views

Sum of the coefficients of the expansion

Find the sum of the coefficients of the expansion: $$\frac{(1+x)\cdot(2+x^2)\cdot(3+x^3)...(103 + x^{103})}{103!}$$ The answer says let $x=1$, is this the way to go? Why not let $x=0$ ??
1
vote
5answers
83 views

$x=yx$. Can this statement be true when we don't know that $y=1$?

I am dealing with an equation which is saying that $yx=x$. On the other hand it is telling us that $\frac{x}{x}=1$ which connotes that $x=x$. Is it not absurd to say that $x=x=yx$ if $x\neq{0}$ and ...
3
votes
0answers
53 views
+50

How to find the inverse arc in the configuration space

The following Figure shows the function from configuration space (Torus) to operational space (Annulus). There is a naturally defined continuous function from configuration space $(\theta_A, ...
1
vote
3answers
32 views

Prove that $a^2+b^2+c^2+d^2+e^2 > a(b+c+d+e)$

Prove that $a^2+b^2+c^2+d^2+e^2 > a(b+c+d+e)$ Seems to be easy but, cannot see the method right now. Tried adding known things like $a^2+b^2>=2ab$ and so on with other letters.Maybe I didn't ...
1
vote
7answers
51 views

Logarithms with an answer that is a fraction

How does log base $16$ of $32$ equal $1.25$? If we divide $32/16=2$ but then if we divide $2/16$ it doesn't come out to a whole number unlike with log base $2$ of $4$ where $4/2=2$ and $2/2=1$ I am ...
1
vote
1answer
24 views

Application of Dimensional Analysis Problem

It is given that the radius $R$, in meters, of the expansion of a liquid in the soil is given by $t$ (time elapsed since the liquid was released), the mass $M$ of the liquid released and of the ...
-1
votes
3answers
36 views

simplify the equation [on hold]

I need help simplifying this equation. It is a fraction just in case the way I formatted it doesn't turn out right. $$ \frac{(4x + 3)^{1/2} − (x + 6)(4x + 3)^{−1/2}}{(4x+3)} $$
1
vote
1answer
34 views

Expected value of prime lottery ticket

Below is a problem I think that I have solved correctly, but cannot seem to get the correct answer. Any help would be greatly appreciated. You pay $\$13.00$ for a ticket. When you buy a ticket, ...
2
votes
2answers
31 views

How to go about solving this inequality question?

$\cos(3x-\pi/3) \leq (1/2).$ Here is what I have done so far... Let $3x-\pi/3 = X$. So I need to solve $\cos(X) \leq 1/2$. Which is all $X$ from $\pi/3$ to $5\pi/3$, so-- $\pi/3 \leq X \leq 5\pi/3 ...
5
votes
4answers
212 views

Why is $\frac{\sqrt{x+1}-1}{x}$ equal to $\frac{1}{\sqrt{x+1}+1}$?

I'm working with the expression $$\frac{\sqrt{x+1} - 1}{x}.$$ According to Wolfram Alpha "alternate form" section (http://www.wolframalpha.com/input/?i=%28%28x%2B1%29%5E1%2F2-1%29%2Fx) it is equal to ...
0
votes
2answers
35 views

To find inverse of function [on hold]

Given $ f(x) = \begin{cases} 2x, & \text{if $x\in[0,1]$} \\ 8 - 2x, & \text{if $x\in [2,3)$} \end{cases} $ Then how to find inverse of f ?
3
votes
1answer
30 views

If $ j , k , n$ are consecutive integers and $jn$ has last digit $9$, what is the last digit of $k$?

$ j , k , n$ are consecutive integers such that $0 < j < k < n$ and the units (ones) digit of the product $jn$ is $9$, what is the units digit of $k$? SAT Question. I don't know if we are to ...
2
votes
0answers
33 views

Periodicity of a sum of periodic functions?

The sum of two periodic functions is periodic if: a) Both periodic functions are continuous b) If the ratio of their fundamental periods is rational Can someone explain why the first ...
1
vote
3answers
34 views

Translate a point on a circumference

If I have a point $A$ on the circumference of a circle with origin $O$ and radius $r$, how would I find the coordinates of point $B$, which is also on that circumference, but is $d$ units away from ...
1
vote
2answers
45 views

solving nonlinear equations

Suppose I have the following two nonlinear (degree two) equations: $y = x^2$ $y = 8 – x^2$ By solving these two equations, the possible values for $x$ and $y$ are: $x = –2, +2$ and $y=4$. Note ...
4
votes
8answers
98 views

factor the following expression $25x^2 +5xy -6y^2$

How to factor $$25x^2 +5xy -6y^2$$ I tried with $5x(5x+y)-6y^2$. I'm stuck here. I can't continue.
0
votes
2answers
25 views

Is this a solution to the equation $a|bx|+c=0$?

I was working on solving a problem in math class, and I was given this problem, $a|bx|+c=0$, as a challenge to solve. This is what I came up with. $$ a|bx|+c=0 \\ a|bx|=-c \\ |bx|=\frac{-c}{a} \\ ...
0
votes
0answers
18 views

Formula for roots of a polynomial, and nature of roots in detail, depending on the discriminant

I am searching for some authentic formula for finding roots of a cubic polynomial, if someone could provide me? I have to solve $$-a r^3 + r^2 - 2 m r + Q^2 = 0$$ for $r$. I am also interested in ...
0
votes
2answers
24 views

Simplifying the expression. [on hold]

Can anyone help with this? You invest $£ 4,624$ in a saver account with an interest rate of $0.42$ percent compounded monthly. After how many months does the balance on you account reach $£ ...
0
votes
2answers
39 views

Simplifying the exponential expression $e^{-4\ln x +8\ln y +2}$ [on hold]

I'm totally stuck on this. Tried numerous sites for a decent explanation but can't find anything. Simplify the expression $$e^{-4\ln x +8\ln y +2}.$$ Thanks in advance.
1
vote
2answers
16 views

Order of Inverse Operations

so this is a very simple question but I am having a tough time with it. So it's finals week and I'm studying up for an Algebra 2 final. The only part I am having trouble with is finding the inverse ...
0
votes
2answers
31 views

Solve for $m$ in $d^m = n$ [duplicate]

I believe the answer is $m = \lceil \sqrt[d]n \rceil$ or $\lfloor \sqrt[d]n \rfloor$. Can anyone help me?