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
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2answers
24 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
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
21 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
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 $a+b$.
0
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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?
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
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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 ...
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
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 ...
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
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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
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, ...
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
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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
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
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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
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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
68 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
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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?
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?
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 ...
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
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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.
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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
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 ...
-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
vote
1answer
110 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
33 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 ...
1
vote
2answers
56 views

solve $\sqrt{x+7}<x$ for $x\in \mathbb{R}$

solve $\sqrt{x+7}<x$ I tried $\sqrt{x+7}<x\\ x+7<x^2\\ x^2-x-7>0\\ x\in \left(-\infty, \dfrac{1-\sqrt{29}}{2}\right) \cup \left( \dfrac{1+\sqrt{29}}{2},+\infty\right) $ I m not ...
-5
votes
3answers
62 views

What's the value of $i^i$? [duplicate]

What's the value of $i^i$?Is it real or imaginary?[$i$ here denotes imaginary number.]
-1
votes
1answer
30 views

Inverse of rational function [on hold]

I need help with this question: Determine whether the given function is one-to-one, and if so, find the inverse: $$ f(x) = 5x + \frac{2}{x} $$ Wolfram says the answer is $\frac{1}{10}\left(x ...
3
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3answers
37 views

solve $|x-6|>|x^2-5x+9|$

solve $|x-6|>|x^2-5x+9|,\ \ x\in \mathbb{R}$ I have done $4$ cases. $1.)\ x-6>x^2-5x+9\ \ ,\implies x\in \emptyset \\ 2.)\ x-6<x^2-5x+9\ \ ,\implies x\in \mathbb{R} \\ 3.)\ ...
0
votes
0answers
17 views

dependent variable change attribution

I have a model that is generally represented as $y = w x z$. Period over period, each of these independent variables will change, and therefore so will the dependent variable. I am trying to ...
-2
votes
1answer
21 views

What are the coordinates of a point given its distance from another point?

If the abscissa of a point is twice the value of the ordinate and has a distance of $2\sqrt{17}$ units from the point $(4,-5)$, what are the coordinates of the point?
0
votes
1answer
26 views

Write all elements of A.A = {$x|x^2<x<10$,x is a whole number}. Answer: A ={$x|x^2+1=0$}.Explain like i'm five.

Write all elements of A.A = {$x|x^2<x<10$,x is a whole number}. Given Answer: A ={$x|x^2+1=0$}. Is this a typo?
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.
1
vote
3answers
149 views

What are the products of real solutions of this equation?

How can I solve $\:\: \log^2_{1/2}(4x)+\log_2\hspace{-0.06 in}\left(\hspace{-0.06 in}\frac{x^2}{8}\hspace{-0.06 in}\right)=8 \;$ ? I have tried the elementary for logarithms simplifying the terms in ...
1
vote
3answers
36 views

Convert the equation to rectangular form $r = \frac {6}{1-\sinθ}$

Convert the equation to rectangular form $r = \frac {6}{1-\sinθ}$ The answer should be: $y = \frac{1}{12} x^2 -3$ But how to arrive at the answer? I tried replacing r with $\sqrt{x^2 + y^2}$, then ...
0
votes
2answers
52 views

Explain the role of the numerator and denominator of a rational exponent such as $\left(\frac{27x^3}{8y^9}\right)^{-\frac{5}{3}}$ [on hold]

So I understand how to solve this problem, (8y^9 / 27x^3)^5/3 ((2y^3)^5 / (3x)^5 (2y^3)^5 / (3x)^5 (32y^15) / (243x^5) but i am confused as to what the direct role of the numerator and denominator is ...