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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0
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0answers
6 views

Confused about proof of division

I thought I was familiar with the regular euclidian algorithm, but I am having trouble understanding a step in this proof from my notes, I am looking for any clarification. $\mathbf{Thereom:}$ Let ...
0
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1answer
16 views

Looking to understand proposition related to the fundamental theorem of algebra

I am having some problem understanding exactly what the following proposition is saying. Also, is this result have a common name? How important it is, etc. It is $\mathbf{Proposition:}$ Let ...
0
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2answers
28 views

Find six triples of positive integers $(a, b, c)$ such that in $ \frac{9}{a} + \frac{a}{b} + \frac{b}{9} = c$.

Solve for $a, b$ and $c$ in the following equation such that Find six triples of positive integers (a, b, c) such that $$ \frac{9}{a} + \frac{a}{b} + \frac{b}{9} = c$$ I have tried various ...
-1
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0answers
13 views

Solve for b and d

Solve for b and d in the following equation. A triangle with sides (a, a, b) has the same area and the same perimeter as a triangle with sides (c, c, d) where a, b, c and d are positive integers ...
2
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0answers
39 views

What is the maximum amount of solutions to $f(x+1)f(x)= ax^2+bx+c$.

$f(x)$ is going to be in the form $mx+h$ thus, $(mx+m+h)(mx+h) = ax^2+bx+c$. With basic algebra $m= \pm \sqrt{a}$. Also $(m+h)(h)=c$. I would guess that because $(m+h)h=c$ has two solutions max if $m$ ...
2
votes
3answers
38 views

Can't determine if given relation is equivalence relation

Definition of relation ~ $(a,b)$ ~ $(c,d)$ $\iff$ $bc^2=da^2$, where $(a,b),(c,d)$ are from $\mathbb{R}\times\mathbb{R}$ and $(a,b),(c,d)$ are different from $(0,0)$ First of all, I wonder if ...
2
votes
4answers
98 views

What is the value of the expression: $(1+\frac 12)(1+\frac 13)(1+\frac 14)…(1+\frac {1}{2004})(1+\frac {1}{2005})$?

What is the value of the expression: $(1+\frac 12)(1+\frac 13)(1+\frac 14)...(1+\frac {1}{2004})(1+\frac {1}{2005})$? This question appeared on the UKMT senior maths challenge 2005, and I can't find ...
0
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0answers
60 views

How to calculate the Minimum of a set of Complex numbers?

Suppose you have 5 complex numbers $$2+4i,\ 6-3i,\ -9-7i,\ -12+23i,\ 3+4i.$$ How do you calculate the Minimum? And does it even make sense? If so, what would be a real world example? Thanks, Shane
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6answers
140 views

If, $x+y=1, x^2+y^2=2$ Find $x^7+y^7=??$

If, $x+y=1, x^2+y^2=2$ Find $$x^7+y^7=??$$ any help guys please?
1
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2answers
19 views

Solving for x using two derivatives and algebra.

There are two things I don't understand about the following: " Set these derivatives equal to each other and solve the resulting equation. $2\sqrt3\cos(x) = 2\sin(x)$ $= \sqrt3 = \tan(x)$ (since ...
4
votes
1answer
18 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
29 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 ...
1
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2answers
58 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 $y$?
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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, ...
3
votes
3answers
53 views

Find the smallest possible value for: $a+b$ [on hold]

If $a,b$ are positive integers with $a, b > 1$, and $$\sqrt{a\sqrt{a\sqrt{a}}}=b,$$ find the smallest possible value for $a+b$.
0
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0answers
49 views

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

If $$x^y=y^x \in\mathbb {R}$$ Find $x,y$. Any help guys?
0
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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
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
129 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
53 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 ...
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 ...
0
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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 ...
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
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 ...
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 $h$.
0
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0answers
36 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, ...
0
votes
2answers
33 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
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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
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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
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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 ...
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 $C$: ...
1
vote
2answers
51 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
2answers
32 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
69 views

If $|z-2|=1$, what are the maximum and minimum values $|z+i|$ can have? [on hold]

If $|z-2|=1$, what are the maximum and minimum values $|z+i|$ can take?
1
vote
2answers
67 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?
3
votes
3answers
100 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?
0
votes
1answer
31 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 ...
0
votes
2answers
41 views

Largest integer $x$ that satisfies $\dfrac{4x+19}{x+5}<\dfrac{4x-17}{x-3}$

Find the largest integral $x$ that satisfies $\dfrac{4x+19}{x+5}<\dfrac{4x-17}{x-3}$ I tried $ \dfrac{4x+19}{x+5} < \dfrac{4x-17}{x-3}\\~\\ (4x+19)(x-3)<(4x-17)(x+5)\\~\\ x<-7 ...
1
vote
2answers
37 views

Quadratic Absolute Value Equation

Problem: Find all $x$ such that $|x^2+6x+6|=|x^2+4x+9|+|2x-3|$ I can't understand how to get started with this. I thought of squaring both sides of the equation to get rid of the modulus sign, ...
1
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3answers
49 views

Finding X from Exponential Equations

$$2^x \cdot 4^{1-x}= 8^{-x}$$ I wrote all the base numbers as a power of 2 but I'm not sure what to do after.
-1
votes
2answers
30 views

Express $x+y+z$ in terms of $a$ and $b$ [on hold]

If $A = X + Y$ and $B = X + Z$, find the value of $X+Y+Z$ in terms of $A$ and $B$.
-3
votes
2answers
62 views

Solving for $x$ in $A=B\cdot \cos(x)+C\cdot \sin(x)$ [duplicate]

I´m working on a little paper, and I want to know if it´s possible in any way to solve this: $$A=B\cdot \cos(x)+C\cdot \sin(x)$$ $A$, $B$ and $C$ are known. I need a way to get the $x$ without using ...
-4
votes
1answer
26 views

Evaluate $\log 64$ using the change of base formula? [on hold]

Is that even possible? I mean, there is no base.
0
votes
1answer
45 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 ...
-1
votes
2answers
60 views

In the equation $2n^3 + 3n^2 = 500,000$, what does $n$ equal? [on hold]

In the equation $2n^3 + 3n^2 = 500,000$, what does $n$ equal?
1
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1answer
111 views

How can one solve $1^x=2$?

Sure, common sense says there's no solution. But, I feel, there should be one! (If there isn't, can't we construct one?)
3
votes
6answers
76 views

How do you show that $\displaystyle\lim_{x\to 0}\frac{\sin(x)}{\sqrt{x\sin(4x)}} $does not exist? [on hold]

How can I show that $\displaystyle\lim_{x\to 0}\frac{\sin(x)}{\sqrt{x\sin(4x)}} $does not exist ?
0
votes
0answers
34 views

Determine the minimum and maximum angles, to the nearest tenth of a degree, that a pipe can make with the horizontal.

For residential drains, a horizontal pipe needs to have a minimum slope of 1/4 inch per foot and a maximum slope of 1/2 inch per foot for waste to drain properly. This means that for every horizontal ...
1
vote
4answers
44 views

Trigonometry equation. Not sure about solution.

The equation goes as follows: $$\sin x +\cos x = 1 + \sin x \cos x$$ and here is how I solved it: $$(\sin x+\cos x)^2=(1+\sin x\cos x)^2$$ $$\sin^2x+2\sin x\cos x+\cos^2x=1+2\sin x\cos ...