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
1answer
5 views

Arrangement of any number of objects from $n$ objects

Prove that the total number of arrangements of objects by taking any number of objects from $n$ different objects is $\lfloor e \times n! - 1 \rfloor$, where $e$ is the natural base. I tried it ...
1
vote
2answers
25 views

Positive roots of polynomial $q(x)=p(x)+k^2$

Let $p(x)$ a polynomial of degree $n\in\mathbb N$ such that $$p(x)=0$$ has exactly $n$ real and positive solutions. Is it true that polynomial $q(x)=p(x)+k^2$, for $k\in\mathbb R$ has only positive ...
17
votes
4answers
160 views
+50

Inclusion-exclusion-like fractional sum is positive?

Let $A_1,A_2,\ldots,A_n$ be finite nonempty sets. Is it true that $$\sum_{i=1}^n\frac{1}{|A_i|}-\sum_{1\leq i<j\leq n}\frac{1}{|A_i\cup A_j|}+\sum_{1\leq i<j<k\leq n}\frac{1}{|A_i\cup ...
0
votes
3answers
707 views

How to calculate cross point $(x, y)$ of circle circumference and $90^\circ$ triangle

I got stuck on this problem, that is rather easy to present but I don't know how to solve it. So I want to get $x$, $y$ coordinates of the point where circle and hypotenuse crosses. Circle radius ...
4
votes
3answers
163 views

Absolute Value inequality help: $|x+1| \geq 3$

Find the solutions to the inequality: $$|x+1| \geq 3$$ I translate this as: which numbers are at least $3$ units from $1$? So, picturing a number line, I would place a filled in circle at the ...
13
votes
5answers
2k views

Prove a number is composite

How can I prove that $$n^4 + 4$$ is composite for all $n > 5$? This problem looked very simple, but I took 6 hours and ended up with nothing :(. I broke it into cases base on quotient remainder ...
1
vote
2answers
12k views

Conversion from binary fraction to octal

Can someone verify my results? Binary to Octal 1) 10101.11 is 25.75 2) 0.01101 is 0.6875 3) 10110110.001 is 266.125
0
votes
0answers
35 views

Considering bank-interest and inflation rates to calculate remaining money in the account

Peter has A [35,000₤] in bank and banks gives B [350₤] per month as interest; he immediately puts C [100₤] back to the to account and spend the rest of it R [250₤] till next months. Every month, ...
1
vote
2answers
56 views

Solving an equation that contains a logarithm

I have the follwing equation: $$y=\frac 1 4x^2 -\frac 1 2 \ln{x}$$ How can $x$ be expressed in terms of $y$?
0
votes
2answers
22 views

Resultant Temperature

Ok im not totally sure if this problem can be solved without the theories of physics; but here goes: With three different unknown quantities x,y and z of the same kind of liquid of temperatures 9, ...
0
votes
4answers
2k views

A tricky logical (age) problem

It's an old question, may be from 7th grade, but I am really looking for a good explanation for this question: A says to B, "I am three times as old as you were, when I was as old as you are". If ...
3
votes
3answers
51 views

Find the smallest possible value for: $a+b$

If $a,b$ are positive integers with $a, b > 1$, and $$\sqrt{a\sqrt{a\sqrt{a}}}=b,$$ find the smallest possible value for $a+b$.
3
votes
2answers
160 views

How to mathematically color the regions bounded by a parametric curve?

Usually, if an implicit equation $F(x, y) = 0$ defines a curve (or curves) on the x-y plane, then we can use the inequalities $F(x, y) < 0$ or $F(x, y) > 0$ to color the regions bounded by the ...
0
votes
0answers
46 views

IF $x^y=y^x$, Find $x,y$ [duplicate]

If $$x^y=y^x \in\mathbb {R}$$ Find $x,y$. Any help guys?
6
votes
5answers
271 views

Given that $x^y=y^x$, what could $x$ and $y$ be?

It's not too difficult to figure out that $x$ and $y$ can both be 1, and also $x$ can be 2 and $y$ can be 4 (and vice versa). But I can't rule out if there are other solutions. Does it have anything ...
5
votes
10answers
277 views
+50

Prove that if $a,b \in \mathbb{R}$ and $|a-b|\lt 5$, then $|b|\lt|a|+5.$

I'm trying to prove that if $a,b \in \mathbb{R}$ and $|a-b|\lt 5$, then $|b|\lt|a|+5.$ I've first written down $-5\lt a-b \lt5$ and have tried to add different things from all sides of the ...
1
vote
3answers
165 views

Trig function phase shift

I've come across a problem and I'm not successful in finding a definition of phase shift that would adress my question. Is the phase shift of $-\cos (x + \frac{\pi}{2})$ equal to $\frac{\pi}{2}$ or ...
0
votes
1answer
44 views

What is the proof for this sum of sum generalized harmonic number?

I believe this sum: $$\sum_{m=2}^k\sum_{n=1}^{m-1}(nm)^{-s}$$ to be equal to $$\frac 12((H_k^{s})^2-H_k^{(2s)})$$ where $H_k^{s}$ is the generalized harmonic number. I only discovered this by ...
0
votes
1answer
44 views

I'm trying to solve for a stopping time given a distance. Think I have the answer.

Trying to work with grouping variables and eliminating the exponent. Please help by explaining how you come to a different answer. The equation is $870t=16t^2$ My logic is to divide $t$ from both ...
4
votes
1answer
96 views

Trigonometric ratio of multiple and sub multiple angles

Given that $a$ lies in 1st quadrant and $$ \sin a +\cos a +\operatorname{cosec} a+\sec a+\tan a+\cot a=7$$ then we have to prove that $\sin(2a)$ is a root of $$x^2-44x-36.$$ I have tried to break all ...
-4
votes
1answer
23 views

How long will it take two clocks to show the same time once again? [on hold]

There are two analog wall clocks on a wall. On 1st January 2000 daytime, John sees the watches through a mirror placed on the opposite wall showing 10:30 A.M. and 1:30 P.M. respectively. The first ...
0
votes
2answers
125 views

Joining two graphs

Suppose I have $f_1(x)=x$ And i restrict its domain as $\color{blue}{(-\infty,0]}$ using $g_1(x)=\dfrac{x}{\frac{1}{2\left(x-0\right)}\left(x-0-\left|x-0\right|\right)}$ Resulting in : Now, ...
3
votes
2answers
52 views

Plot of $y=x+0\sqrt{-x}$ (and WolframAlpha vs Desmos)

To plot the graph of $y=x+0\sqrt{-x}$ : Do we have to first find out the domain of $y$ which is $y \in ( -\infty,0 ]$ ? $\color{blue}{\text{[Case 1]}}$ (that's what I do) Or do we solve the ...
9
votes
6answers
6k views

How can you find the cubed roots of $i$?

I am trying to figure out what the three possibilities of $z$ are such that $$ z^3=i $$ but I am stuck on how to proceed. I tried algebraically but ran into rather tedious polynomials. Could you ...
1
vote
2answers
66 views

Find conditions for $a$ and $b$ such that $P(x)=x^4-(a+b)x^3+(ab+2)x^2-(a+b)x+1$ has only real roots.

I need to find conditions for a and b such that $$P(x)=x^4-(a+b)x^3+(ab+2)x^2-(a+b)x+1$$ has only real roots. Any hints on how I should do that?
0
votes
0answers
146 views

Solve system of equations for the ratios of the vectors

(Sorry for the bad title, didn't think of a better way to describe the problem). I have a system $\mathbf{A}\in\mathbb{C}$ that forms the problem $\mathbf{Ax}=\mathbf{b}$, for which I want to find an ...
9
votes
4answers
730 views

If a function can only be defined implicitly does it have to be multivalued?

What is the general reason for functions which can only be defined implicitly? Is this because they are multivalued (in which case they aren't strictly functions at all)? Is there a proof? ...
4
votes
1answer
70 views

Find the remainder when the sum is divided by $1000$

Find $S \pmod{1000}$ given: $$S = \sum_{n=0}^{2015} n! + n^3 - n^2 + n - 1$$ $$S_0 = 0! + 0 - 0 + 0 -1 = 0$$ $$S_1 = 1! + 1 - 1 + 1 - 1 = 1$$ $$S_2 = 2! + 8 - 4 + 2 - 1 = 7$$ This isn't ...
0
votes
1answer
10 views

Constructing exponential function using a table of outputs

I have been given the exponential function $g(x)=ar^{x}$. I have also been given the table $(x=4,g(x)=\frac{256}{3})$, and $(x=5,g(x)=\frac{1024}{9})$.... Now as far as I understand you can take ...
0
votes
2answers
18 views

Use algebra to decide the rectangle's area

I understand that with the usage of variables, I can use algebra to come up with the right area for the blue rectangle. So I let all the different sides be different variables. Now I know that I ...
1
vote
2answers
493 views

What annual installment will discharge a certain debt?

What annual installment will discharge a debt of $ 717.60 due in 4 years at 20% p.a. simple interest, if the installments are paid at the each end of each year? I tried the following: $ 717.60 ...
0
votes
3answers
64 views

Problem with simplifying $\frac{(3+h)^2-9}{(3+h)-3}$ [on hold]

I need help simplifying $$ {(3+h)^2-9\over (3+h)-3}. $$ The answer is $6+h$. I keep getting $h$.
3
votes
8answers
100 views

If $f(x)=4x^2+ax+a-3$ is negative for at least one negative $x$ find all possible values of $a$

If $f(x)=4x^2+ax+a-3$ is negative for at least one negative $x$ find all possible values of $a$ I don't know how to find all possible values. I tried making the lower of the two roots as ...
4
votes
4answers
47 views

finding $a_1$ in an arithmetic progression

Given an arithmetic progression such that: $$a_{n+1}=\frac{9n^2-21n+10}{a_n}$$ How can I find the value of $a_1$? I tried using $a_{n+1}=a_1+nd$ but I think it's a loop.. Thanks.
0
votes
2answers
32 views

Finding the parameter a [on hold]

The ratio of the roots of the equation $x^2 +ax + a+2=0$ is $2$ Find the values of parameter $a$. I don't understand what the question means .
1
vote
2answers
33 views

solution of an algebraic equation?

There is an algebraic equation like $ax^{2n-2}-bx^{2n-4}+c=0$, where $a,b,c>0$ and $n$ is an integer with $n\geq3$. What are the solutions of this equation or the properties of its solutions?
0
votes
1answer
19 views

Average rate of change help.

A function is given. Determine the average rate of change of the function between the given values of the variable. $f(x) = 2 − x^2 $ $x = 8, x = 8 + h$ I solved for $f(8)$ and got $-62$... I ...
1
vote
2answers
13 views

Function to apply to a linearly increasing positive real number to reach an arbitrary limit

I've got a friend who is making a browser game and he's trying to figure out how to make a function that acts like a logarithm in that it returns higher values quickly but eventually mellows out and ...
11
votes
6answers
457 views

Elementary proof that $\pi < \sqrt{5} + 1$

I wanted to show that $$ \frac{\pi}{4\phi} < \frac{1}{2} $$ Where $\phi$ is the golden ratio. I have confirmed the results numerically, and by simple algebra the inequality simplifies down to $$ ...
11
votes
5answers
5k views

How can I write an equation that matches any sequence?

One thing I have been wondering about lately is how to write an equation that describes a pattern of numbers. What I mean is: x 0 1 2 y 1 5 9 If ...
0
votes
2answers
74 views

Write the Linear equation [duplicate]

In 1940 there were $245,300$ immigrants admitted to a country. In 2006 there were $1,060,431$. Write a linear equation expressing the number of immigrants, $y$, in terms of $t$, the number of years ...
19
votes
7answers
1k views

An oddity in some linear equations

Okay, so I've started Algebra I this year, and i've always had a love for math. And at one point in the course we were presented with an equation similar to this one: $5x + 3 = 8x + 3$ And so I ...
2
votes
1answer
210 views

Algebra Word Problem

If I drove $38.91$ miles, and my car gets $22$ miles to the gallon, and gas cost $ $3.699$ per gallon, how much did I spend on gas? I'm sure this is an algebra problem, but I can't think where to ...
1
vote
1answer
42 views

What is the sum of all $k$ values?

In an urn there are a certain number (at least two) of black marbles and a certain number of white marbles. Steven blindfolds himself and chooses two marbles from the urn at random. Suppose the ...
0
votes
1answer
45 views

Help ! What is the equation?

I have $2$ Variables: Job ($A, B, C$) Age (Young, Adult, Old) Total population for job is $100$, total population for age is $100$ Job $A$ has $20\%$ of population Job $B$: $30\%$ Job $C$: ...
0
votes
1answer
68 views
3
votes
3answers
98 views

Solve $(x+1)^n=(x-1)^n$, assuming $x$ is a complex number and $n>0$.

How do I solve $(x+1)^n=(x-1)^n$? I assumed $x=a+bi$, getting the equation $((a+1)+bi)^n=((a-1)+bi)^n$. How do I solve it using Moivre's n-th root theorem?
1
vote
2answers
50 views

values of sin of multiples of 10? [on hold]

I was in class the other day and the professor was arguing that sin(1), sin(10), and sin(100) are all equal to the same value and that calculators are incorrect due to approximations. This problem has ...
-2
votes
3answers
27 views

Determine $P(x)$, with real coefficients and the lowest possible grade, such that $0$, $1+i$ and $1-i$ are its roots and $P(-2)=1.$ [on hold]

I need to determine $P(x)$, with real coefficients and the lowest possible grade, such that $0$, $1+i$ and $1-i$ are its roots and $P(-2)=1.$ How can I solve this problem?
0
votes
1answer
29 views

Do I need to use different trig functions in different quadrants?

I don't have any formal education in Trigonometry or Calculus, but I'm studying a book on Pre-calc before school begins this fall. I've completed College level Algebra too, so math isn't something ...