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
votes
0answers
10 views

ratio of derivatives

({deriv(y, x)}^7derivN(x, y, 3)+{deriv(y, x)}^3derivN(y, x, 3))/({deriv(y, x)}^2{derivN(y, x, 2)}^2) When I looked at this problem I thought the problem is incorrect as we have not been given y as a ...
0
votes
1answer
18 views

What is the greatest remainder if you divide a 2-digit number by its digit sum

I just found this problem and tried to solve it. I wrote x=90a+b and tried to maximize the function f(a,b)=(9a+b)/(a+b) but did not come to any solution. Then I considered 10a+b = x (mod a+b) which ...
2
votes
2answers
39 views

Algorithm for multiplying infinite decimals?

What is the (best) algorithm for multiplying two real numbers based on their decimal expansions? Obviously the algorithm can't be completed but I mean an algorithm that will successively approximate ...
1
vote
3answers
31 views

What is the value of xyz?

If $a,b$ and $c$ are not equal to $0$ and $1$ and if $a^x=b,b^y=c,c^z=a$,then $xyz=?$ We have tried to solve by equation,but it can't produce the desired result.
1
vote
2answers
24 views

Pairs $x_i+x_j$ positive for total positive sum

Let $x_1,\dots,x_n\in\mathbb{R}$ be such that $x_1+\dots+x_n>0$. At least how many sums $x_i+x_j$ ($i<j$) must be positive? It is possible that $x_1=n$ and $x_2=\dots=x_n=-1$, in which case $n-...
1
vote
3answers
31 views

Calculating integer solutions of a logarithmic equation

The question asks: Calculate the integer solutions of the equation $\log_2(x+2)+\frac12\log_2(x-5)^2=3$ To me, this is trivial if solved in the following way: $(x+2)+(x-5)=2^3$ $2x-3=8$ Answer: $x=\...
1
vote
1answer
28 views

Simplifying $\frac{^n\mathrm{C}_k}{^n\mathrm{C}_{k-1}}$

Question asks to simplify: $$\frac{^n\mathrm{C}_k}{^n\mathrm{C}_{k-1}}$$ I have a few steps but not sure if its correct. $$\begin{align*}\frac{(n)!}{(n-k)!(k)!} \bigg/ \frac{(n)!}{(n-k-1)!(k-1)!}...
0
votes
2answers
38 views

What is the coefficient of $x^{2m}$ in $(1 + 4 x - 2 x^2 + 4 x^3 + x^4)^m$?

For each positive integer $m$, write $(1 + 4 x - 2 x^2 + 4 x^3 + x^4)^m = \sum_{j = 0}^{4m} b_j^{(m)} x^j$. What is $b_{2m}^{(m)}$ in terms of $m$?
-3
votes
2answers
44 views

simplify factorials: $\frac{(k-1)!}{(k+2)!}$ [on hold]

Question: simplify $$\frac{(k-1)!}{(k+2)!}$$ What I did was: $$\frac{(k - 1)!k!}{(k + 2)! \cdot (k + 1)!}$$ This I did following the rule $n! = n \times (n - 1)!$. can this be simplified ...
0
votes
2answers
57 views

Simplifying factorials: $\frac{(n-1)!}{(n-2)!}$

Question: simplify $$\frac{(n-1)!}{(n-2)!}$$ What I did was: $$\frac{(n - 1)!}{(n - 2)! \times (n - 3)!}$$ This I did following the rule $n! = n \times (n - 1)!$. But my answer just doesn't look ...
0
votes
1answer
12 views

Divide items with integer ID-s into N equal groups, based on ID-s

I have unknown number of items, each having ID (consecutive integer numbers), ie. 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15... I want to split above items into as ...
0
votes
1answer
29 views

Finding a rule for the outcome of $\sqrt[4]{r}(a+\sqrt{r})$

It is well known that $$\sqrt{4+3\sqrt{2}}=\sqrt[4]{2}(1+\sqrt{2})\tag{1}$$, and similarly, $$\sqrt{10+6\sqrt{5}}=\sqrt[4]{5}(1+\sqrt{5})\tag{2}$$$$\sqrt{6+4\sqrt{3}}=\sqrt[4]{3}(1+\sqrt{3})\tag{3}$$$$...
0
votes
1answer
22 views

Hoses in the Road [on hold]

Perhaps you have seen the hoses in the road measuring traffic flow and vehicle speed. A car traveling over them will first hit the back hose and then the front hose. A computer measures the time ...
0
votes
2answers
25 views

Finding the domain and range of a difficult piecewise composite function

I recently inquired about finding a formula for a composition of two piecewise functions, but I have been thoroughly confused by a slightly different example. In this case, I have a question about ...
1
vote
1answer
24 views

Need help with inductive proof of Binomial Theorem

I'm new to math and trying to learn about the Binomial Theorem, by following this tutorial. I got stuck trying to read the Induction Proof. They give an example of using the Sum notation: $$ (x + y)^...
0
votes
2answers
30 views

finding out total digits in a large number

Is there any easy way to find out how many digits does the number $12^{400}$ have or such types of problems like how many digits the number $x^y$ have? ($x$ and $y$ are variables)
0
votes
2answers
25 views

Simplfying complex rational expression

I'm trying to simplify $$ \frac {\dfrac {x}{y} - \dfrac {y}{x}}{y}.$$ My method of trying to solve this is try to simplify the numerator $\frac {x}{y}-\frac{y}{x}$ Then I find the GCD: $xy$, multiply, ...
5
votes
1answer
98 views

Find all solutions to $f\left(x^2+xf(y)\right)=xf(x+y)$

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f\left(x^2+xf(y)\right)=xf(x+y)$$ for all $x,y\in\mathbb{R}$. This is somewhat related to this question, but with an $xf(y)$ term instead ...
3
votes
2answers
48 views

Simplifying trig expression $\frac{1}{1-\cos \theta}$

I need help with the following trig problem, I'm getting the first part, but can't seem to complete it. $$\frac{\cos \theta}{1-\cos^2 \theta}- \frac{1}{1-\cos \theta}$$ The first part is going to ...
0
votes
1answer
31 views

Express $c$ and $d$ in terms of $m$ where $c$ and $d$ are zeroes of $f$ where $m > -2$

Let $$f(x) = x^2 - mx -(6m^2+25m+25)$$ where $m > - 2$ It can be shown that $f(x)$ has two zeroes. Suppose we have $c,d \in \mathbb R$ s.t. $c < d$ and $f(c) = f(d) = 0$, express $c$ and $d$ ...
5
votes
4answers
397 views

Finding a tricky composition of two piecewise functions

I have a question about finding the formula for a composition of two piecewise functions. The functions are defined as follows: $$f(x) = \begin{cases} 2x+1, & \text{if $x \le 0$} \\ x^2, & \...
-2
votes
3answers
18 views

Convert ranges to other ranges

It may be pretty basic but I am looking for an efficient way to convert a number with a range of 15-40 to another number in the range of 255-0. so at ...
4
votes
1answer
41 views

Maximum and minimum of $f(x)=\cos(\sin(x))-\sin(\cos(x))$

Given the function: $$f(x)=\cos(\sin(x))-\sin(\cos(x))$$ it has absolute maxima at $x=(2k+1)\pi$ with $k=0,1,..N$ and relative maxima at $x=2k\pi$. It is not clear where are the minima. Putting the ...
0
votes
0answers
46 views

analytical solution to equation

I am trying to solve the following equation by $x$. (put sole $x$ on one side of the equation), but I am not sure if there is a analytical solution to this problem: $$\large -2\ln\left( \frac{(1-p)^{...
1
vote
3answers
49 views

Find all the angles $v$ between $-\pi$ and $\pi$

Find all the angles $v$ between $-\pi$ and $\pi$ such that $$-\sin(v)+ \sqrt3 \cos(v) = \sqrt2$$ The answer has to be in the form of: $\pi/2$ (it must include $\pi$) I have tried squaring but I get ...
-5
votes
1answer
41 views

Why does the modulo affect other terms in the equation? [on hold]

i just want to ask if why does the modulo affect the other terms in an eqution? Why does the 4th equation has to be multiplied by $a^2$? Then as the modulo becomes $n≡1(mod3)$ in the 5th eq. then the ...
0
votes
1answer
37 views
2
votes
1answer
35 views

On the expression $(2x+1)$ of odd numbers

I have this problem: Find three odd consecutive numbers with the property that the product of the first one and the third one minus the product of the first one and the second is greater by eleven ...
1
vote
2answers
37 views

If $(ax+b)(x-3)=4x^2 + cx - 9$ for all values of $x$, what is the value of $c$?

I am confused with this problem. Things get confusing because in quadratic form $b$ is actually called $c$ in this problem. Also because I'm not sure where to start. Because it says for all values of $...
0
votes
0answers
27 views

Dirichlet Kernel function: Can you please derive the solution for $n=0$ on the Cosine equation? [on hold]

Below is the link for the intro of Dirichlet function: http://i.stack.imgur.com/ydWab.png I came across an equation $$\frac{1}{2} + \cos{t} + \cos{2t} + \dots + \cos{nt} = \frac{\sin((2n+1)\frac{t}{2}...
2
votes
3answers
45 views

if $x \neq 0$ and $x = \sqrt{4xy - 4y^2}$, then how does expressing $x$ in terms of $y$ mean $x = 2y$?

I have the equation $x=\sqrt{4xy - 4y^2}$, and I know that $x=2y$ when expressed in terms of $y$, but I'm not sure of the process to get there. I know that \begin{align}\sqrt{4xy - 4y^2} &= \...
0
votes
1answer
28 views

Correct Order of Applying Graphical Transformation with Absolute Value

I was going through this website, reading about transformations of graph when $| |$ is applied to various parts of a given function, $y=f(x)$. Going through the fourth example of the page, I came ...
0
votes
2answers
23 views

Calculating length of vertical line bisecting parallel arcs

I have 2 arcs, offset from one another (never intersecting) and a vertical line through them both (NOT at the center of the arcs). Is there a way to calculate the vertical distance between the 2 arcs? ...
0
votes
4answers
53 views

If $x$ is positive, then why does $\frac{1}{\sqrt{x+1} + \sqrt{x}} = \sqrt{x+1} - \sqrt{x}$?

Given that $x$ is positive, $\frac{1}{\sqrt{x+1} + \sqrt{x}} = \sqrt{x+1} - \sqrt{x}$ I've been trying to convert the left side of the equation to the right side: $$ \frac{1}{\sqrt{x+1} + \sqrt{x}}$...
0
votes
5answers
46 views

If $4x^2 + 9y^2 = 100$ and $(2x + 3y)^2 = 150$, then why is the value of $6xy = 25$?

I know that $4x^2 + 9y^2 = 100$ and $(2x + 3y)^2 = 150$ I've converted $(2x + 3y)^2 = 150$ to be $$ 2x^2 + 12xy + 3y^2 = 150$$ Which I can then convert to $$ 12xy = 150 - 2x^2 - 3y^2$$ I'm ...
0
votes
0answers
35 views

Math precalculus/trig

Circle $O$ below has radius 1. Eight segment lengths are labeled with lowercase letters. Six of these equal a trigonometric function of $\theta$. Your answer to this problem should be a six letter ...
14
votes
2answers
384 views

Sum of roots rational but product irrational

Suppose that $x_1,x_2,x_3,x_4$ are the real roots of a polynomial with integer coefficients of degree $4$, and $x_1+x_2$ is rational while $x_1x_2$ is irrational. Is it necessary that $x_1+x_2=x_3+x_4$...
4
votes
2answers
63 views

If $5\sqrt [ x ]{ 125 } =\sqrt [ x ]{ { 5 }^{ -1 } } $, then $x$ equals $-4$

For the equation $$5\sqrt [ x ]{ 125 } =\sqrt [ x ]{ { 5 }^{ -1 } } $$ $x$ is equal to $-4$, but I'm not sure why. I've taken the right side of the equation ${ \left( \frac { 1 }{ 5 } \right)...
1
vote
3answers
32 views

If $p$ and $q$ are odd, prove that there are no integral solutions

Prove that if $p$ and $q$ are odd numbers , then the equation $x^{10} + p x^{​9} + q = 0$ does not have integral solution. Could some hint a simple approach to solve this question. I am not getting ...
1
vote
4answers
93 views

If $n$ is a positive integer, then $(-2^n)^{-2} + (2^{-n})^2 = 2^{-2n+1}$

I'm not sure why $$(-2^n)^{-2} + (2^{-n})^2=2^{-2n+1}$$ I have been going over this equation for a while now, noticing, and have successfully got quite far in the equation, finding that $$ (-2^n)^{-...
3
votes
4answers
71 views

Solving Quadratic system of equations

Solve this system of equations: $$(1) \quad 0=-10x^2-9xy+50x-25y-7y^2+5$$ $$(2) \quad 0=-5x^2-17xy+25x+50y-14y^2+7$$ Shame on me but I'm failing to solve this system. I can't see a short (...
0
votes
2answers
37 views

Find equation for $f(x)=a^{-x}$ graph from points given?

An example of the graph I want to find the equation for this graph with the known point of (120,5)and approximate points of (300,2) (225,2.5) and (150,4). Can I do this or do i need more information?...
1
vote
2answers
51 views

Is $2\prod_{i=1}^{n}{p_i} - \prod_{i=1}^{n}{(p_i + 1)} - \prod_{i=1}^{n}{(p_i - 1)}$ even and negative for $n > 1$?

Is $$2\prod_{i=1}^{n}{p_i} - \prod_{i=1}^{n}{\left(p_i + 1\right)} - \prod_{i=1}^{n}{\left(p_i - 1\right)}$$ even and negative for $n > 1$, where $p_i > 1 \hspace{0.07in} \forall i \in \left[1,n\...
1
vote
2answers
56 views

Set of $n$ numbers

Let $1<q\le2$. We have a set of $n$ numbers $1,q,q^2,...,q^{n-1}$. Prove that set of numbers can be divided into $2$ parts, so that the sum of numbers in parts not differ by more than $1$. Let $1&...
2
votes
3answers
92 views

Proving that $x^{16} > 5$ when given a polynomial of degree $15$. [on hold]

I am unable to prove the following If $x^{15}-x^{13}+x^{11}-x^9+x^7-x^5+x^3-x = 7$ prove that $x^{16} > 15$.
2
votes
1answer
58 views

Simplify this equation.

Can I simplify or approximate this equation without sigma and combination? \begin{align} \sum_{i = 0}^n (-1)^i {n \choose i} \frac{{d+1}}{d(di + 1)} \end{align}
-2
votes
0answers
16 views

Intersection of Circles and Triangulation [on hold]

Tracking a Cellphone CT1 to CT2 = 700m CT2 to CT3 = 1200m CT1 to CT3 = 1350m Cell Phone is 600m from CT1, 650m from CT2, and 800m from CT3 Draw a circle in each Cell Tower, indicating the distance ...
3
votes
3answers
169 views

On linear equations and equations with fractions

I'm learning how to solve equations. When solving linear equations I've learnt we have 2 elemental and allowed operations, one of them is the following: If $P=Q$ is an equation then $aP=aQ$ is an ...
0
votes
1answer
14 views

Forming a expression of quadratic equation involving polygons

Six congruent isosceles triangles with equal sides $x$ cm are removed from the six corners of a paper in the shape of a regular hexagon of sides 20cm . The remaining portion is in the shoe of a 12 ...
-1
votes
0answers
19 views

Absolute Value Graph Problem in Gelfand's Functions and Graphs

I am working through Gelfand's Functions and Graphs, where I am currently on the absolute value section. At the end of the chapter practice problems, Gelfand poses a set of problems regarding ...