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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1answer
37 views

are there closed form solution to $n \cdot y + \log(y) = x$?

I am trying to find a closed-form solution to $n \cdot y + \log(y) = x$ How do we deal with the fact that there aren't, if there aren't? Is it possible to rewrite this in a better way as a ...
0
votes
4answers
112 views

Which number is bigger? [duplicate]

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 ...
-1
votes
3answers
52 views

Trigonometric double identities [on hold]

Prove the following trigonometric identity: $$(\sec (x) - 1)(\sec (x) + 1)=\frac{\sin^2(x)}{\cos^2 (x)}$$
0
votes
1answer
13 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 ...
-7
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1answer
34 views

I am at lost… please help [on hold]

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
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6answers
66 views

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

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
21 views

Short structured question [on hold]

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
40 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
23 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
45 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
16 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
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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
42 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
2answers
375 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 ...
7
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 < ...
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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 ...
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votes
0answers
36 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
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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
74 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
782 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
66 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
127 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
50 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
27 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