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
0answers
39 views

Impossible System of Equations

This is from a competition: DMM Olympiad, Ural State University P4 I don't understand what the question means exactly (the first part, i.e. "exclude $x$ or $y$ from..." part). Does it mean "write $x$ ...
20
votes
2answers
233 views

How to prove $\displaystyle\sum_{n=0}^{\infty} \dfrac{1}{1+n^2} = \dfrac{\pi+1}{2}+\dfrac{\pi}{e^{2\pi}-1}$

How can we prove the following $$\sum_{n=0}^{\infty} \dfrac{1}{1+n^2} = \dfrac{\pi+1}{2}+\dfrac{\pi}{e^{2\pi}-1}$$ I tried using partial fraction and the famous result $$\sum_{n=0}^{\infty} ...
0
votes
6answers
127 views

Solve: If $3a=2b$, then what is the value of $\frac{3a-b}{2a+b}$? [closed]

If $3a=2b$, then what is the value of $\dfrac{3a-b}{2a+b}$?
1
vote
1answer
38 views

The speed of learning and prior

If I know $$\frac{\alpha}{\alpha+\beta}<\frac{\lambda}{\lambda+\gamma}$$ can I know the sign of $$\frac{\alpha+1}{\alpha+1+\beta}<\frac{\lambda+1}{\lambda+1+\gamma} $$ And the sign of ...
-1
votes
1answer
65 views

give direct proof of the fact $a^2 - 5a + 6$ is even for any integer [duplicate]

I know this is true but I don't know how to prove it. I have worked it out for the integers from $1$ to $10$ but this is not direct proof, is there a formula I need?
1
vote
1answer
63 views

Find sum of the roots of quadratic polynomials [closed]

The zeroes of a quadratic polynomial $x^2+ax+b$ are $c$ and $d$ and the zeroes of a quadratic polynomial $x^2+cx+d$ are $a$ and $b$. Find the value of $a+b+c+d$. The thing doesn't make sense how ...
4
votes
6answers
77 views

Solving $\dfrac{x+2}{x}>0$

I want to find values of $x$ such that $\dfrac{x+2}{x}>0$ : $1+\dfrac{2}{x}=\dfrac{x+2}{x}>0 \implies \dfrac{2}{x}>-1 \implies \dfrac{1}{x}>\frac{-1}{2} \implies x<-2 $. But by ...
2
votes
0answers
40 views

Zeilbergers algorithm in Maple

I try to prove several hard combinatorial identities. One of them is following \begin{align*} \sum_{s=0}^{\min\{k,n-1\}} \sum_{i=0}^{k-s} (-1)^{i} {2n+k-i-1 \choose k-s-i} {i-n \choose s} {n+i-1 ...
1
vote
3answers
86 views

Which number is higher $2^{600}$ or $3^{400}$?

Which number is higher $2^{600}$ or $3^{400}$ ? I know that the solution is $3^{400}>2^{600}$ bot how to explain that. without using a calculator.
-1
votes
2answers
15 views

finding the ratio which divides the segment

Find the ratio in which the point (2,-1) divides the segment from (6,1) to (0,-2). Find the coordinates of the point that divides the segment from (0,-1) to (6,3) I the ratio 2:5 . can somebody ...
0
votes
1answer
23 views

How many sets of 2 without duplicates out of these options?

So there are twelve signs of the zodiac: Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricorn, Aquarius and Pisces I want to know how many possible sets of 2 I can make ...
-3
votes
0answers
43 views

a question in calculus

Let $a,b,c$ be positive integer numbers such that \begin{align*} &1\leq ab,\\ & \frac{1}{b}\leq c \leq a. \end{align*} Is the following relation correct? $$\frac{ab+1}{2\sqrt{abc}}\leq ...
3
votes
2answers
76 views

Square root of $x^3$

I understand the concept behind the expression $\sqrt{x^2} = |x|$. So, then why is the square root of $x^3$ NOT equal to $|x|\sqrt{x}$? Specifically, I can write $\sqrt{x^3}$ as $\sqrt{x^2\times ...
0
votes
1answer
31 views

Find the domain of square ln function [closed]

The question is to find the domain of this function: $$f(x)=\sqrt{\ln \dfrac{x-4}{x+2}} + \sqrt{4-3x-x^2}$$ I don't really know where to start, I couldn't find any example on Google like this.
-2
votes
1answer
49 views

Calculate $S =\sum_{k=1}^n\frac {1}{k(k+1)(k+2)}$. [duplicate]

Calculate $S =\sum_{k=1}^n\frac {1}{k(k+1)(k+2)}$. I know I posted this question already but I want a more detailed answer. For example, how you got from one step to another using the partial fraction ...
4
votes
3answers
87 views

Solve $x^4-3x^2+1=0$ in terms of cosine.

I put the equation in the form of a quadratic: $(x^2)^2-3x^2+1=0$ Then using the quadratic formula, $x^2=\frac{3\pm\sqrt{9-4}}{2}$ $x^2=\frac{3+\sqrt{5}}{2}$ and $\frac{3-\sqrt{5}}{2}$ ...
2
votes
2answers
58 views

Show that $\frac{(n-a)^2}{n}$ can be written as $1-\left(\frac{n}{a}\right)^2\cdot\frac{n}{(n/a)^2}$

Hi I need to show that $\frac{(n-a)^2}{n}$ can be written as $1-\left(\frac{n}{a}\right)^2\cdot\frac{n}{(n/a)^2}$. I have got so far to $(a^2/n)-2a+n$ But I can not see how to proceed. Can anyone ...
2
votes
5answers
125 views

Why does $\sum\limits_{i=1}^n i^2 = An^3+Bn^2+Cn + D$?

I've got this question because of this video (around 3:15). I wonder how setting up a system of $3$ equations will help him solve this problem. I'm thinking I might not understand this because I've ...
3
votes
4answers
103 views

Calculate $ S =\sum_{k=1}^n\frac {1}{k(k+1)(k+2)}. $

Calculate $S =\displaystyle\sum_{k=1}^n\frac {1}{k(k+1)(k+2)}$. This sequence is neither arithmetic nor geometric. How can you solve this. Thanks!
3
votes
2answers
34 views

Example of Parseval's Theorem

In the textbook "Mathematics for Physics" of Stone and Goldbart the following example for an illustration of Parseval's Theorem is given: Until 2.42 I understand everything but I don't understand ...
-1
votes
1answer
26 views

proportion question [closed]

A contractor agrees to lay a road $3000$ metres long in $30$ days. $50$ men are employed and they work $8$ hours a day. After $20$ working days he finds that only $1200$ metres of the road is ...
-1
votes
1answer
17 views

inverse proportion question [closed]

if $y$ is inversely proportional to $2x+1$ and the difference in the values of $y$ when $x=0.5$ and $x=2$ is $0.9$, find the value of $y$ when $x=-0.25$ . please provide workings
0
votes
3answers
49 views

Proving the trigonometric identity $4\sin\left(x + \frac{\pi}{6}\right)\sin\left(x - \frac{\pi}{6}\right) = 3 - 4 \cos^2 x$

I am using the addition and difference formulae for cosine and sine but seem to be getting stuck... somewhere. Prove that $$4\sin\left(x + \frac{\pi}{6}\right)\sin\left(x - \frac{\pi}{6}\right) ...
2
votes
2answers
39 views

Convert the Polar Equation to Cartesian Coordinates

$$ r^2=\sec 4\theta $$ I graphed this equations using Wolfram Alpha and found it to be 2 hyperbolas. I'm having difficulty showing this using the standard equations $$ x=r\cos\theta \;, \; ...
-2
votes
3answers
51 views

Question about Ellipse [closed]

Given an ellipse, with the center at the origin and a given value of $16$ as its "$a$". There is also a point $(8,6)$ on the ellipse. The "$b$" value is less than "$a$" (so it is a wide ellipse). How ...
0
votes
3answers
42 views

What does the notation $\min_x$ mean?

I have a problem in which I need to find $\min_x(f(x))$. What does this notation mean?
5
votes
4answers
193 views

How can we say two algebraic expressions are “equal” if one is undefined at certain points and the other isn't?

I'm trying to understand why it is that we can say $\frac{x^2-1}{x-1} = \frac{(x-1)(x+1)}{(x-1)} = x+1$ but then have it also be the case that the two functions $f(x) = \frac{x^2-1}{x-1}$ and ...
1
vote
3answers
41 views

Set of Numbers with GCD equal to $1$

Can someone give me a set of $4$ positive integers with $3$ of them having a common divisor that is greater than $1$, but the GCD of all four positive integers is $1$.
1
vote
2answers
22 views

Proof of the Greatest Denominator [duplicate]

Can someone help me prove this. It is a proof of the sum and difference of the greatest common denominator. Given: x and y are integers with a GCD of 1. Prove: that the GCD of x + y and x − y is ...
1
vote
0answers
65 views

Why is the “i” disappearing?

The task is: Find the argument in its simplest form. $$(\sin(x) +i(1-\cos(x)))^2$$ where $x$ is an acute angle. I multiplied out the equation and let alpha be the required argument, then ...
3
votes
1answer
37 views

What is the remainder when polynomial $f(x)$ is divided by $(x+1)(x-3)$ when $f(-1) = -4$ and $f(3) = 2$?

A polynomial $f(x)$ gives remainder $2$ when divided by $(x-3)$ and gives a remainder $-4$ when divided by $(x+1)$. What is the remainder when $f(x)$ is divided by $(x^2 - 2x - 3)$? I have shortened ...
0
votes
2answers
37 views

Graph of $y=\text{constant}*x$

Graph of $y=x$ , $y=\frac{1}{2}x$ and $y=\frac{1}{4}x$ is All are straight line. But with different slope. Why?
1
vote
1answer
26 views

Fourth Order Homogeneous Ordinary Differential Equation With Double Complex Conjugate Roots (2.10-14)

This is actually a problem in algebra as shall be seen. I need to find the general solution for the following differential equation: $$y''''+8y''+16y=0$$ The characteristic equation for this is: ...
3
votes
2answers
51 views

In the formula $Ax+By=C$, is it true that $A$ and $B$ can't both be zero? If so, why not?

I read in a math book that in the formula $Ax+By=C$, I read that $A$ and $B$ can't both be zero. I think C will also be zero because anything times zero equals zero and on a graph, the x- and y- ...
2
votes
1answer
85 views

Extremely hard problem with absolute values [closed]

I need help with this problem. If $a$ and $b$ are positive integers such that $$|a-1|+|a-2|+|a-3|+\cdots+|a-2015|=b(b+1)$$ find the sum $a+b$.
2
votes
2answers
58 views

Find a, b, c if three equations are given?

I was given three equations in term of $a, b$ and $c$. Equations are as follows $ab (a+b+c)=1001$ $bc(a+b+c)=2002$ $ac(a+b+c)=3003$ Find $a, b, c$. MY ATTEMPT I took tue ratio and I got relation as ...
4
votes
1answer
97 views

Inequality in 4 variables

I came across the following problem in a book. Four real numbers $p,q,r,s$ satisfy $p+q+r+s=9$ and $p^2+q^2+r^2+s^2=21$. Prove that there is a permutation $a,b,c,d$ of $p,q,r,s$ Such that $ab-cd\ge ...
1
vote
1answer
42 views

If something is $2^{N+1}$, how can I get $N$ back from the end result?

I need this for a GIF encoder I'm programming, if something is $2^{N+1}$, how can I get $N$ back from the end result? For example, $2^{7+1} = 256$, how can I get back to $7$ from $256$? I've spent ...
0
votes
3answers
122 views

A symmetric inequality with three variables

Here is an inequality I came across in a book that I was doing:- Prove that for all $a,b,c\gt 0$ $$\frac {a+b+c}{(abc)^{1/3}}+\frac {8abc}{(a+b)(b+c)(c+a)}\ge 4$$ I have no idea about how to approach ...
0
votes
1answer
18 views

A tough rearranging

In a physics textbook I try to follow some math derivation, and got stuck here: First we rearrange Eq. (30.13) to the form $$\frac{di}{i-(\mathcal{E}/R)}=-\frac{R}{L}dt$$ And equation (30.13) ...
0
votes
1answer
60 views

Equation with logarithms $\lg{2}+\sqrt{\lg(\lg{x})}=\lg{(x+10)}$

Let be the following equation: $$\lg{2}+\sqrt{\lg(\lg{x})}=\lg{(x+10)}$$ The solution belongs to the following interval: $(10^{?}, 10^{?})$. Which is the form of the interval? Thanks!
0
votes
2answers
21 views

How to calculate what percentage a fraction is

So I have the following: X/.9125 I want to turn this into X * Y where Y equals a percentage. So for example, 500/.9125 = 547.95 rounded. I would like to ...
0
votes
2answers
36 views

Simple Algebra to solve

Could I get a walk through solving the problem: 100 = Y - (Y x 0.0875) I started by subtracting Y on both sides: ...
0
votes
2answers
22 views

Trapezium problem

I am trying to solve the following geometry exercise: In an isosceles trapezium the sum of its bases is equal to $6\sqrt{2}$ cm and the minor base is equal to the half of the major base. Suppose the ...
5
votes
1answer
117 views

How can one find intermediate digits of a root of an algebraic equation?

I was wondering whether there is a way to find intermediate digits of an algebraic equation. For example, if I have $$234x^{\frac{1}{12345}}-24621x^{\frac{1}{3456}}=1$$ And I want to find the ...
1
vote
3answers
46 views

Are these exponential forms equal?

Is $(\frac1{\sqrt x})^{11}$? the same thing as $x^{\sqrt{11}}$ ? Basically what I'm asking is are those equivalent/the same?
0
votes
4answers
56 views

Finding the X- Intercepts for this Equation

How do you find the x intercepts for this problem? (Please show and explain how you solved it) $$y= x^3-10x^2+31x-30$$
0
votes
2answers
36 views

graphing a trig function to find its limit.

I am just starting to learn about limits and am trying to solve the problem $\lim_{\theta\to 0}f(x) = \sin (2\theta)/\theta$ I thought the best way to solve this would be to substitute values that ...
2
votes
3answers
46 views

Solve $2x-1 \le x^2$ for $x$

My book tells me that the solution is $(-\infty, \infty)$. But why is that the solution if you get $(x-1)^2 \ge 0$ after you finish factoring the equation? Shouldn't the answer be $(1,\infty){}{}{}$? ...
0
votes
2answers
41 views

How to factorize a long expression without long division?

I have this: $3n^{3} + 16n^{2} + 23n + 10$. I need to decompose it in (n+1)(n+1)(3n + 10). Using long division I can do it, but I was wondering, in order to save space on a long proof I'm writing ...