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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1answer
31 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 ...
-1
votes
1answer
37 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
42 views

values of sin of multiples of 10?

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 ...
-3
votes
3answers
23 views

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

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
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1answer
61 views

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

If $|z-2|=1$, what are the maximum and minimum values $|z+i|$ can take?
1
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2answers
47 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
85 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
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1answer
24 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
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2answers
37 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
36 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
44 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
29 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 $B$.
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votes
2answers
59 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 ...
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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
28 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 ...
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votes
2answers
59 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
108 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
74 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
31 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
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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
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2answers
55 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 ...
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votes
3answers
61 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.]
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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
votes
3answers
36 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 ...
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votes
1answer
20 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
3answers
39 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
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3answers
147 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
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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 ...
2
votes
3answers
56 views

Least Common Denominator: $ \frac{\sqrt{x}}{x}+\frac{\ln\ x}{2\sqrt{x}} $

$$ \frac{\sqrt{x}}{x}+\frac{\ln \ x}{2\sqrt{x}} $$ I have tried combining these two fractions; however, I keep getting stuck. $$\frac{2\sqrt{x}}{2\sqrt{x}}\cdot\frac{\sqrt{x}}{x}+\frac{\ln\ ...
0
votes
0answers
35 views

What is the difference between the scalar and vector components of a vector?

What is a scalar component of a vector and what is a vector component of a vector. suppose a vector is making and angle theta with the origin then in my book it is written that its x component is the ...
1
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2answers
80 views

Proving inequalities using Calculus

In general how do you prove inequalities using calculus, I believe it is using maxima or minima right? For example $$a^2b+b^2c+c^2a \le 3, \qquad a,b,c \ge 0,\quad a+b+c=3.$$ How would you use ...
3
votes
0answers
41 views

Solving for a variable in an inverse function

I was asked to solve this formula for $R_2$: $$\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$$ So I did the following: \begin{align*} \frac{1}{R_2} &= \frac{1}{R} - ...
0
votes
3answers
60 views

Solve $3^{2x} -2 \cdot 3^{x+5} + 3^{10} = 0$ for $x$

Here's the question: Solve for $x$ in $$3^{2x} - 2 \cdot 3^{x+5} + 3^{10} = 0$$ I know that you have to factor something out, I'm just not sure what that something is. Thanks in advance
4
votes
2answers
34 views

Solve for $x$ in: $e^{2\ln(x)-\ln(x^2+x-3)} = 1$

So the question is to solve for x in: $$e^{[2\ln(x)-\ln(x^2+x-3)]} = 1$$ I took the natural log of both sides, and simplified. Here is what I've gotten it down to: $$2\ln(x) = \ln(x^2+x-3)$$ And I'm ...
1
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1answer
19 views

Arithmetic, Geometric and Harmonic prove equation.

If $a, b, c, d$ be in Arithmetic Progression, $a, e, f, d$ be in Geometric Progression, and $a, g, h, d$ in Harmonic Progression respectively; prove that $ad=ef=bh=cg$.
3
votes
1answer
35 views

Solve $x^2-|5x-3|-x<2,\ \ x\in \mathbb{R} $

Solve $x^2-|5x-3|-x<2,\ \ x\in \mathbb{R} $ I tried $x^2-|5x-3|-x<2$ , case $1$ , $x^2-(5x-3)-x<2,\ x\geq 0 \\ x^2-6x+1<0 \\ 3-2\sqrt2 < 3+2\sqrt2 \\ 0.17<x<5.8\\ $ ...
4
votes
2answers
62 views

How do i find $\tan(\theta)$ such that : $\frac{16}{\sin^6(\theta)} + \frac{81}{\cos^6(\theta)}=625$??

How do i find $\tan(\theta)$ such that :$$\frac{16}{\sin^6(\theta)} + \frac{81}{\cos^6(\theta)}=625$$? Note : i used some trigono-form but sorry i didn't succed . Thank you for any help.
0
votes
1answer
37 views

Functions of modulus

How do I calculate the range of any modulus function? I know that if $x <2$ then it's expansion is negative and if $x>2$, it's expansion is negative, but will it help? Consider an example, $$f ...
0
votes
4answers
51 views

Is there any notation for general $n$-th root $r$ such that $r^n=x$?

As we know that the notation for the $n$-th principal root is $\sqrt[n]{x}$ or $x^{1/n}$. But the principal root is not always the only possible root, e.g. for even $n$ and positive $x$ the principal ...
0
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3answers
41 views

Express the quantity as a single logarithm: $ \frac 13 \ln (x+2)^3 + \frac 12[ \ln x - \ln (x^2+3x+ 2)^2]$

Express the quantity as a single logarithm: $ \frac 13 \ln (x+2)^3 + \frac 12[ \ln x - \ln (x^2+3x+ 2)^2]$ The answer in the book is ln $\frac {\sqrt{x}}{x+1}$ If am not allowed to to cancel terms ...
5
votes
1answer
63 views

Function equation, find the function evaluated at the certain point.

Let $f(x)$ be a polynomial with real coefficients such that $f(0) = 1,$ $f(2)+f(3)=125,$ and for all $x$, $f(x)f(2x^{2})=f(2x^{3}+x).$ Find $f(5).$ The constant term, $a_0 = f(0) = 1$. Let: ...
4
votes
4answers
84 views

$(x+y+z)^3-(y+z-x)^3-(z+x-y)^3-(x+y-z)^3=24xyz$?

The question given is Show that $(x+y+z)^3-(y+z-x)^3-(z+x-y)^3-(x+y-z)^3=24xyz$. What I tried is suppose $a=(y+z-x),\ b=(z+x-y)$ and $c=(x+y-z)$ and then noted that $a+b+c=x+y+z$. So the ...
3
votes
8answers
96 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 ...
-3
votes
1answer
37 views

Use the graph of Y=f(x) shown below to answer the following questions

https://www.flickr.com/photos/134404416@N03/shares/6p9h7f I tried answering some of the questions as you can see in the picture link provided, but I just am not sure if my answers are even right. ...
1
vote
2answers
78 views

How to show this fraction is not an integer

Suppose $k\geq 2$ is an integer. I want to show $$\frac{1+k+k(k-2)}{1+\frac{k-1}{k}+\frac{(-1-\sqrt{k-1} )^2}{k(k-2)}}$$ is not an integer. It is equal to $$\frac{(k-2) k (k^2-k+1)}{2 (k^2-2 ...
0
votes
3answers
31 views

How do you determine the end behavior of a rational function?

Example $$\frac{6x + 2}{x^2 - 9} = \frac{6x + 2}{(x + 3)(x - 3)}$$ I know how to find the vertical and horizontal asypmtotes and everything, I just don't know how to find end behavior for a ...
0
votes
2answers
36 views

Find the sum of the roots of the exponential equation

The equation $$2^{333x - 2} + 2^{111x + 2} = 2^{222x + 1} + 1$$ has three real roots. Find their sum. I'll simplify it first as: $$\frac{1}{4}2^{333x} + (4)2^{111x} = (2)2^{222x } + 1$$ Let ...