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
1answer
13 views

Solve the equation below

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
26 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
11 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 ...
2
votes
3answers
43 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
31 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
24 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
37 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
290 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 ...
4
votes
6answers
560 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
17 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 ...
0
votes
0answers
21 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
43 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
60 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
175 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
775 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
62 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
36 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
65 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) = ...
5
votes
0answers
77 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
25 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
32 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
25 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
54 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
32 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
62 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 ...
-6
votes
1answer
26 views

coordination questions 123 [on hold]

A ray of light passing through the point $(1, 2)$ reflects on the x-axis at point A and the reflected ray passes through the point $(5, 3)$ find the coordinates of A. Kindly solve full question.
3
votes
1answer
111 views

Find the number $n^{2}$ from the number $\large n^{n^{n^{2}}}$

Find the number $n^{2}$ from the number $\large n^{n^{n^{2}}}$ Any help? I tried with $\log$ but I got nothing.
0
votes
2answers
20 views

Simple algebraic manipulation with 2 equations

My first equations is this: $ d_2 = d - 30.$ My second equations is this: ${1\over d_2 }= {1\over12} - {1\over1.066(d-30)}$ I am trying to solve for $d_2$ in the second equation and then set the ...
-3
votes
0answers
27 views

Determination of polynomial values [on hold]

The polynomial $R(x)=x^4 + Ax^3 + Bx^2 + 10x-1$ ($A,B \in I$) has a remainder of $-15$ when divided $x+1$ and a remainder of $39$ when divided by $x-2$. Determine $A$ and $B$.
1
vote
2answers
70 views

Ways of coloring the $7\times1$ grid (with three colors)

Hints only please! A $7 \times 1$ board is completely covered by $m \times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the ...
0
votes
1answer
48 views

Why is the discriminant of the discriminant negative?

On this link is a question about functions. My question is, in that question itself, a pivotal part of the solution is to realise that the discriminant of the (positive) discriminant is negative. ...
2
votes
1answer
48 views

Let $ f: R \to [\frac{1}{2} , 1]$ and $f(x+2) = \frac{1}{2} +\sqrt{f(x) -f(x)^2}$

Problem : Let $ f: R \to [\frac{1}{2} , 1]$ and $f(x+2) = \frac{1}{2} +\sqrt{f(x) -f(x)^2}$ Then which of the following is always true $(a) f(2) = f(7)$ $(b) f(4) = f(10) $ $(c) f(2) =f(4) $ ...
0
votes
0answers
21 views

How to derive the relation $T=S_1$ for the equation of a chord of an ellipse whose midpoint is given? [on hold]

How to derive the relation $T=S_1$ for the equation of a chord of an ellipse whose midpoint is given as ( x1, y1 ) ? where T = x(x1)/aa + y(y1)/bb - 1 and S1 = (x1)(x1)/aa + (y1)(y1)/bb - 1 where 2a ...
0
votes
2answers
31 views

why dividing a number by 1.25 gives back 20 percent less of original?

So i had to takeout the discount from price. price = 10 discount = 20% my default method has been: price - price*discount ...
3
votes
1answer
56 views

Rewriting $|x-10|+|y-5|\leq 7$ so that absolute values disappear - Algebra

Equation 1: $|x-10|+|y-5|\leq 7$ I want to rewrite this equation into equations that do not have the absolute value. $|A|\leq b$ can be written as $A \leq b$ $A \geq -b$ I want to apply the ...