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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2
votes
2answers
38 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 ...
2
votes
1answer
101 views

Does such a function exist: $\left|f(x) - f(x+1)\right| \ge {1\over x+1}$?

Does a function $f(x)$ exist such that $\left|f(x) - f(x+1)\right| \ge {1\over x+1}$ where the range of the function does not extend infinitely in the positive direction. If not why?
-6
votes
1answer
66 views
-3
votes
3answers
34 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 $an + b$.
1
vote
0answers
74 views

Compound Interest Formula with regular deposits

I am trying to create a Java Program that can calculate compound interest with an initial deposit and regular deposits (daily, weekly, monthly, quarterly, bi-yearly, and yearly). I'm fairly sure I can ...
0
votes
3answers
47 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
31 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
1answer
41 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 $n$.
1
vote
1answer
43 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 ...
1
vote
1answer
63 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? ...
73
votes
21answers
20k views

Is $.999999999… = 1$?

I'm told by smart people that $0.999999999\ldots = 1$ and I believe them, but is there a proof that explains why this is?
3
votes
1answer
30 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} ...
0
votes
1answer
19 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
1answer
18 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 ...
1
vote
7answers
54 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 ...
-2
votes
1answer
25 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
2answers
27 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 ...
0
votes
1answer
31 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 : ...
1
vote
1answer
56 views

Solve $z^3 + 5z^2 + (9 - 5i)z + 10 - 10i = 0$ [duplicate]

Solve $$z^3 + 5z^2 + (9 - 5i)z + 10 - 10i = 0$$ I have never dealt with equations with complex numbers in them so this is interesting; first Ill expand. $$ \implies z^3 + 5z^2- 5iz + 9z + 10 - 10i = ...
3
votes
1answer
27 views

Why does this method of solution for this system of equations yield an incorrect answer?

We are required to solve the following system of equations: $$x^3 + \frac{1}{3x^4} = 5 \tag1$$ $$x^4 + \frac{1}{3x^3} = 10 \tag2$$ We may multiply $(1)$ by $3x^4$ throughout and $(2)$ by $3x^3$ ...
5
votes
1answer
60 views

Algebra on a Louvre tablet

Problem: On a Louvre tablet of about 300 B.C. are four problems concerning rectangles of unit area and given semiperimeter. Let the sides and semiperimeter be $x,y$ and $a$. Then we have ...
1
vote
1answer
25 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 ...
-4
votes
1answer
37 views
-4
votes
0answers
32 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 ...
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
35 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
0
votes
2answers
30 views

Radius of curvature and continuous functions

Let $\kappa (x)$ be radius of curvature function for a continuous function $f(x)$. Is it necessary that $\kappa(x)$ will have extrema when $f(x)$ does. And the nature of extrema is opposite to that ...
-1
votes
2answers
128 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
40 views

Area of rectangle.

When a quantity is proportional to two or more quantities then why take their product equal to the quantity to which are they proportional?
0
votes
1answer
39 views

Formal Method for Determining the Domain of Solutions to an Equation?

     I'm doing an algebra review packet in order to prepare to take an independent-study calculus class. About five eighths of the way through the problems posed by this ...
2
votes
1answer
42 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
35 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 $x=0$ ??
11
votes
1answer
208 views

Prove that $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+…+\frac{1}{199}-\frac{1}{200}=\frac{1}{101}+\frac{1}{102}+…+\frac{1}{200}.$

I want to prove this high school algebra problem. prove that $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{199}-\frac{1}{200}=\frac{1}{101}+\frac{1}{102}+...+\frac{1}{200}.$$ Please help me. ...
4
votes
8answers
100 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.
1
vote
3answers
33 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 ...
2
votes
2answers
33 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 ...
1
vote
3answers
37 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
1answer
36 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, ...
-1
votes
3answers
36 views

simplify the equation [closed]

I need help simplifying this equation. It is a fraction just in case the way I formatted it doesn't turn out right. $$ \frac{(4x + 3)^{1/2} − (x + 6)(4x + 3)^{−1/2}}{(4x+3)} $$
5
votes
4answers
218 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
39 views

To find inverse of function [closed]

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 ...
1
vote
2answers
48 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 ...
2
votes
0answers
35 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 ...
9
votes
9answers
1k views

examples of functions with vertical asymptotes in real life

As a math teacher, I tend to get the class involved by finding real-life applications of the math- with functions and vertical asymptotes I am having trouble finding simple enough (rational) functions ...
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} \\ ...
8
votes
4answers
637 views

Proof of inequality $e^x + e^{-x} \leq 2e^{x^2}$

How would I prove that $$e^x + e^{-x} \leq 2e^{x^2}, \quad \text{for all real $x$}?$$ I narrowed it down to proving for $x \in (-1,1)$. I observed that for $(0,1)$ and for $(-1,0)$ I may need to ...
1
vote
2answers
231 views

Determine if $\frac{k-1}{k}+\frac{1}{k(k+1)}=\frac{k}{k+1}$ holds

How to prove if the following equality holds? $$\frac{k-1}{k}+\frac{1}{k(k+1)}=\frac{k}{k+1}$$ Maybe finding a common denominator would work, but I have no idea how to do it in this example. I see ...
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 ...