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)

0
votes
4answers
71 views

Which number is bigger?

Which number is bigger? $1.01^{101}$ or $2$? and how about $e^{\pi}$ or $\pi^e$? Tried some algebraic manipulations to no end, so would love some suggestions or some different ways to approach those ...
0
votes
1answer
9 views

convert price per PU in price per ton

Quite basic calculation, but my client insist on telling I am calculating wrong while I am quite sure that my calculation is correct. Given the following variables: weight in kg per packing unit ...
-4
votes
1answer
28 views

I am at lost… please help

A boy skis 4 miles down a mountain slope in 10 minutes. What is his average speed, in miles per hour (mph), over that time interval?
0
votes
8answers
57 views

The sum of two positive integers is 35 and their product is 304. What is the lesser number?

The sum of two positive integers is 35 and their product is 304. What is the lesser number? Can anyone show me a step by step tutorial on this?
-3
votes
1answer
20 views

Short structured question

After Gary donated 1/3 of his pocket money and Terry donated 1/2 of his pocket money, Gary has \$29 more than Terry. The amount donated by Gary is \$8 more than that donated by Terry. How much ...
2
votes
2answers
39 views

simplifying complex fractions

I have this expression ${2 \over x^2 - 4}$ + ${1 \over x + 2}$ So I would take the LCM to ${x^2 - 4}$ or using the difference of 2 squares to be ${(x + 2)(x - 2)}$ So I would simplify this to: ${2 ...
1
vote
1answer
30 views

How to factor a third degree polynomial? [on hold]

How do I factor $2x^3-6x$? I've looked it up, but I haven't seen advice that helps with my exact problem.
2
votes
5answers
30 views

Is it possible to factor a quadratic equation when $a$, $b$, and $c$ are all equal?

I have the equation $4x^2+4x+4$ to factor. I know that need to start with $$(2x \quad )(2x \quad )$$ to make $4^2$, but I can't seem to factor the rest of the way. What should I do?
0
votes
1answer
21 views

Solving function notations. [on hold]

A gas station attracts customers by offering coupons worth a 0.05 discount for every $1.00 spent on gasoline. a. Complete the table. The equation that represents the relation, using the ...
-1
votes
0answers
45 views

A book on functions! [on hold]

I am looking for a good book on algebra which covers functions very comprehensively. It should cover functions of the type: linear: $f(x) = ax + b$ quadratic: $f(x) = ax^2 + bx + c$ cubic: $f(x) = ...
-2
votes
1answer
43 views

Solve the equation below [on hold]

Solve the equation $$\tan(\cos^{-1}\sqrt{x})=2^{\log_{4}x}.$$ I have no idea where I have to start; it's a little hard for me. So any help?
-2
votes
1answer
28 views

A basic factorial question type [on hold]

Hello could you show me a way that how to solve this kind of questions? a and b are natural numbers $60! = a6^b$ What is the biggest value of b?
0
votes
0answers
15 views

Exchange Rate scenario. Which is the better option?

You are travelling overseas. If you use your ATM or Credit Card, you will be assessed a 3% surcharge on all transactions. So, you decide to withdraw Euros at your local bank branch to see if you ...
3
votes
3answers
104 views

what is wrong with this natural log conversion

why can't we convert this: $$4e^{1+3x}-9e^{5-2x}$$ to this: $$(1+3x)\ln(4)-(5-2x)\ln(9)$$ or this: $$(1+3x)\ln(4)+(5-2x)\ln(-9)$$ this comes from the q, solve: $$4e^{1+3x}-9e^{5-2x}=0$$ do the above ...
1
vote
2answers
41 views

Seating people in a circular table

It has always been an interesting question. If we have $10$ chairs and a round table, how many ways are there of seating $10$ people? I would say there are $10!$ ways to seat the people due to ...
-1
votes
2answers
30 views

Does the order in a circular arrangement matter?

I posted a question a while ago: Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least three adjacent chairs. My question here is: imagine a ...
1
vote
2answers
41 views

( Logarithmic Equation ) Solve for x.

$(x+1)^{log(x+1)} = 100(x+1)$ Attempt at solution : $$ (x+1)^{log(x+1)} = 100(x+1)$$ $$= x^{log(x+1)} + 1 = 100x +1$$ $$=(x+1)+1=100x+1$$ $$=−98=99x$$ $$x=−98/99$$ But the answer given in the ...
2
votes
3answers
371 views

Stars and Bars vs PIE

I randomly made up this question so I could check: There are $3$ kids and $6$ gifts, how many ways to distribute so that each kid has at least one gift. Obviously, $**|**|**$ there are ...
6
votes
6answers
1k views

How to solve this exponential equation?

I'm given this equation and i have to solve for the x. $$ e^{2x} -(e^5 + e^2)e^x + e^7 = 0 $$ The results should be $x =2$ and $x = 5$. At first i thought it would be an easy task, ...
0
votes
0answers
26 views

Stock Dividends

I just do not get it, and the teacher is teaching a class in the other room. Here is the question: Danielle owned 100 shares of FPIC stock. She purchased 100 shares of FPIC on November 14, 2011, ...
1
vote
1answer
18 views

Logarithmically bounded function fulfills $f(n) \le \lceil m \cdot \log_b r \rceil$ for certain numbers $n,m,r$

Let $f : \mathbb N \to \mathbb N$ be a function such that $f(n) \le 1 + \log_b n$ for some base $b$ and all $n$. Now let $n \in \mathbb N$ have the property that $$ \frac{r^m - 1}{r-1} \le n < ...
0
votes
0answers
16 views

What would be the equation for “3% in 100 Hz range, 0.5% in the 2000 Hz range”

The smallest distinguishable pitch/frequency by a human ear is something like this: Pitch is our perceptual interpretation of frequency. As mentioned, ideal human hearing ranges from 20 to 20,000 ...
-1
votes
0answers
35 views

polar coordinate transformation

If we have an equation $\mathcal{L_I}=\prod \mathrm{exp}\bigg(-\lambda_j \displaystyle\sum\limits_{m=1}^{\Psi_{j}}\binom{\Psi_j}{m} ...
-1
votes
1answer
45 views

Modulus Problem [on hold]

I do not understand how to solve such a question: $$|x+1| - |x| + 3|x-1| -2|x-2| = x+2.$$ How would you go about all the possibilities with which sign the modulus could take? Appreciate any help!
1
vote
1answer
62 views

How does one use the complex plane to solve this problem?

Given: $$a^2 + ab + b^2 = 1 + i$$ $$b^2 + bc + c^2 = -2$$ $$c^2 + ca + a^2 = 1$$ Find $$(ab + bc + ca)^2.$$ The solution says to use the complex plane. Can somebody explain to me (an average ...
-3
votes
4answers
179 views

Find the value of the question below [on hold]

If $x^{3}+\frac{1}{x^{3}}=14$ Find the value of $$x^{6}+\frac{1}{x^{6}}$$ Original Question: If $x^{2}+\frac{1}{x^{2}}=14$ Find the value of $$x^{5}+\frac{1}{x^{5}}$$
-1
votes
2answers
73 views

Evaluate the infinite radical expression $2\sqrt{2\sqrt[3]{2\sqrt[4]{2\sqrt[5]{2 \cdots}}}}$ [on hold]

Find the value of $$2\sqrt{2\sqrt[3]{2\sqrt[4]{2\sqrt[5]{2 \cdots}}}} .$$ I really don't know where I start, so any help will be appreciated.
3
votes
3answers
779 views

How can I try myself to solve exponential equations easily?

I spent hours trying to solve: $$4^x + 1 = 2^{x+1}$$ Can you guide me on how to solve this? How can I train myself to always find the right "trick" to solve such equations? Rather than just ...
3
votes
1answer
65 views

How to find unkown height of triangle without hyptenuse

I been trying to solve this question and have tried to solve it for many days, but do not know how, any help would be much oblidged. A cable company owns the roads marked with the dotted lines in ...
-1
votes
0answers
37 views

Puzzle on multiplying by fixed values to reach a target number.

So, this one's tricky. There's a keycode combination, and there are six buttons. Each button multiplies the base number of 1 by their respective multipliers (see below). Once the result number gets ...
2
votes
3answers
67 views

Why would the cubic have $5$ roots?

The polynomial $P(x)$ is cubic. What is the largest value of $k$ for which the polynomials $Q_{1}(x) = x^{2}+(k-29)x-k$ and $Q_{2}(x) = 2x^{2}+(2k-43)x+k$ are both factors of $P(x)$? $P(x) = ...
6
votes
1answer
116 views

Find the sum of the series below

Find the sum $$(1\cdot2)+(1\cdot3)+(1\cdot4)+\cdots+(1\cdot2015)+(2\cdot3)+(2\cdot4)+\cdots+(2\cdot2015)+\cdots+(2014\cdot2015)$$ What I have tried... We are looking for ...
3
votes
2answers
44 views

Find the least $N$ so there is no square

Find the least positive integer $N$ such that the set of $1000$ consecutive integers beginning with $1000 \cdot N$ contains no square of an integer. Let $x^2$ appear before $1000N$ so: $(x+1)^2 ...
1
vote
0answers
26 views

simplifying complex expression

Hi I am trying to simplify the following expression:$$ \left|\frac{1}{a+ib}\left(\frac{J_1(c x)}{J_1(c b)}-x\right)\right|^2,\quad a,b,x\in \mathbb{R}, \ c\in \mathbb{C} $$ Is there a simple way of ...
0
votes
0answers
21 views

Proof: Condition that two quadratic functions may have a common linear factor. [on hold]

Find the condition that two quadratic functions of $(x,y)$ called $ax^2 + bxy + cy^2 $and $a'x^2 + b'xy + c'y^2$ may have a common linear factor.
1
vote
4answers
33 views

Slope of a line segment.

If $A(x_1, y_1)$ and $B(x_2, y_2)$, we know that slope $m = \frac {(y_2 - y_1)} {(x_2 - x_1)}$. What decision can we take aout the line segment when, $m = \frac 0 0$, $m = \frac {dy} 0$, and, $m = ...
0
votes
2answers
55 views

How many divisors of the combination of numbers?

Find the number of positive integers that are divisors of at least one of $A=10^{10}, B=15^7, C=18^{11}$ Instead of the PIE formula, I would like to use intuition. $10^{10}$ has $121$ divisors, ...
1
vote
1answer
62 views

Solving $\frac{9a^3-7ab^2+2b^3}{3a+2b}=3a^2-2ab-b^2+\frac{4b^3}{3a+2b}$

I have the following problem: $$\frac{9a^3-7ab^2+2b^3}{3a+2b}$$ The solution in the book is $$3a^2-2ab-b^2+\frac{4b^3}{3a+2b}$$ but I do not know how to get there. I could solve the other ...
2
votes
2answers
25 views

Show that any 2D vectors can be expressed in the form…

(a) Show that any 2D vector can be expressed in the form $s \begin{pmatrix} 3 \\ -1 \end{pmatrix} + t \begin{pmatrix} 2 \\ 7 \end{pmatrix},$ where $s$ and $t$ are real numbers. (b) Let $u$ and $v$ be ...
1
vote
1answer
49 views

Two infinite radicals question

Hello I have stucked with theese two questions: $\sqrt{a:\sqrt{a:\sqrt{a: \cdots}}} + \sqrt[3]{a\cdot\sqrt[3]{a\cdot\sqrt[3]{a\cdots}}} = 12$ $a=\text{ ?}$ ...
3
votes
4answers
34 views

Manipulating equations question

In the equation: $$T = 2\pi \sqrt {\frac lg}$$ it is for determining period of pendulum swing If I want to solve for $g$ and I want to start by removing the root do I square everything in the ...
1
vote
1answer
37 views

How to use Principle of Inclusion-Exclusion here?

A while ago I posted a question: Coloring a Grid. Online, I seem to have stumbled upon a usage of PIE AOPS Wiki Solution AIME II #9. (1) Now, I have experience with PIE, but I do not see how to ...
0
votes
4answers
85 views

$2n=n^2$ what are the solutions

I have just thought of this and I know someone must have before but is the only solution (with real numbers) to $2n=n^2$ $n=2$
1
vote
1answer
30 views

Why is $2^4$ congruent to $-1$ modulo $17$?

I saw an interesting question on Quora (What remainder is obtained when $2^{2017}+1$ is divided by $17$?), but I do not understand the author's solution: Three, because $$ \begin{align} 2^{2017} ...
0
votes
1answer
26 views

Converting complex query to algebra

Algebra is all I know, and I cannot resolve this using my rudimentary algebra. What is the correct procedure? On Wednesday all items at a clothing store were $15. Brenda bought a number of ...
-1
votes
1answer
27 views

Algebra with two variables [on hold]

I have tried answering the following question using algebra, but as it has two variables, I don't know how to solve. Please advise the correct procedure. Sandra had short and tall glasses in the ...
0
votes
0answers
36 views

Find the distance between two points

I'm learning on my own and having some problems understanding these 2 exercises: 1) $$ d = \sqrt{(-\sqrt{6}-\sqrt{3})^2+(0-(-\sqrt{5}))^2} $$ $$ = \sqrt{6+2\sqrt{18}+3+5} = \sqrt{14+2\sqrt{9*2}} ...
-3
votes
3answers
56 views

Find the sum of all products of two distinct naturals, neither exceeding 2015. [on hold]

Find the sum $$(1\cdot2)+(1\cdot3)+(1\cdot4)+\cdots+(1\cdot2015)+(2\cdot3)+(2\cdot4)+\cdots+(2\cdot2015)+\cdots+(2014\cdot2015)$$ any help? I tried with telescope but got nothing
-3
votes
0answers
35 views

solve another system of three equations [on hold]

I have: $x=\dfrac{-.5b-.5c+.25d}{b+c+d}$ $y=\dfrac{.5b\sqrt{3}+.5c\sqrt{3}+.25d\sqrt{3}}{b+c+d}$ $z=b+c+2d$ I need help moving the $b$, $c$, and $d$ to the Left-hand-side; and moving the x, y, and ...
1
vote
0answers
64 views

Solve a system of three equations [on hold]

$x=\dfrac{a-.5c+.25d}{a+c+d}$ $y=\dfrac{.5c\sqrt{3}+.25d\sqrt{3}}{a+c+d}$ $z=a+c+2d$ How do I make it so that only an $x$, an $y$, and/or and $z$ are on the Right-Hand-Side of the equation while ...