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
68 views

Converting $x^y-y^x=1$ Into $y=$ Form

This is probably going to be a simple answer, but, how would you convert $x^y-y^x=1$ into $y=$ form without any $y$ on the opposite side of the equation?
2
votes
4answers
66 views

Solve for $v$ - simplify as much as possible

Solve for $v$. Simplify the answer. $$-3 = -\frac{8}{v-1}$$ Here is what I tried: $$-3 = \frac{-8}{v-1} $$ $$(-8) \cdot (-3) = \frac{-8}{v-1} \cdot (-8) $$ $$24 = v-1$$ $$25 = v$$
0
votes
2answers
20 views

How do I solve this system using graphing?

$y_A$ and $y_B$ represent continuous linear relations. Some values from the relations are shown in the table below. \begin{array}{|c|c|c|} \hline x & y_A & y_B \\ \hline -8 & -5 & ...
0
votes
3answers
50 views

Graphing systems of linear equations.

I'm currently finishing the unit systems of linear equations and I ran into trouble while attempting to read the the table of values. I am able to graph systems of equations and find solutions on a ...
0
votes
1answer
30 views

Calculating a Break Even point using Marginal Cost

Calculus: I am asked to determine the break even point given the following units. Fixed Cost: \$$3000$, Marginal Cost: \$$1$, Selling Price: \$$3$ I know how determine break even from the standard ...
2
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3answers
60 views

Proof of $a^n>n$ by non-analytic method

Is there any proof of the fact that $a^n>n$ for all sufficiently large $n$ where $2>a>1$, without using methods from calculus?
2
votes
2answers
81 views

Prove the root is less than $2^n$

A polynomial $f(x)$ of degree $n$ such that coefficient of $x^k$ is $a_k$. Another constructed polynomial $g(x)$ of degree $n$ is present such that the coefficeint of $x^k$ is $\frac{a_k}{2^k-1}$. ...
5
votes
2answers
54 views

Evaluate $\sum_{r=0}^n \binom{n}{r}\sin rx \cos (n-r)x$

Evaluate $$ \sum_{r=0}^n \left[\binom{n}{r}\cdot\sin rx \cdot \cos (n-r)x\right] $$ I tried to use binomial identities, but since there are trigonometric terms, I don't have the idea ...
0
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1answer
51 views

Combinatoric meaning of multinomial coefficients

$$\binom{n}{k}$$ means how many ways there are to choose $k$ objects from $n$ total objects. What is the combinatoric meaning of: $$\binom{n}{k_1, k_2, ... , k_n}$$ ??
1
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1answer
33 views

The properties of the sum of exponentials

I have the following equation: $$P > \sum \limits_{i=1}^ n \exp(a_i\cdot t)$$ where $a_i \in[0,1]$. I'm trying to find $t$. I realize that there's no simple way to get the log of a sum, but I was ...
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2answers
67 views

Solve $4^{x-1}+2^{x-2} = 68$ [closed]

Please help solving $4^{x-1}+2^{x-2} = 68$.
0
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1answer
28 views

Domain and range of $y=f(7)$

My answer: The only real number in the x-coordinate is $7$. Then, Domain ={7} Every real number in the y-coordinate is paired with $7$. Then, Range $=\mathbb{R}$ But Wolframalpha answer is: Domain ...
2
votes
2answers
67 views

On the inner workings of induction?

I always had some doubts on the inner workings of induction. So I decided to make a little experiment. I am familiar with the proof that the sum of the first $n$ integers is $\cfrac{n(n+1)}{2}$ so I ...
2
votes
2answers
48 views

Finding values of $a$ with which two equations are equivalent; getting rid of radical sign

Two equations are given: $$x^2+(a^2-5a+6)x=0$$ $$x^2+2(a-3)x+a^2-7a+12=0$$ We need to find the values of $a$ that will render them equivalent. From the first equation, $$x=-a^2+5a-6$$ From the ...
4
votes
1answer
118 views

Determine all functions $f:\mathbb{Q}\to\mathbb{Q}$ satisfying the functional equation $f(2f(x) + f(y)) = 2x + y$

Determine all functions $f$ defined on the set of rational numbers that take rational values for which $$f(2f(x) + f(y)) = 2x + y \tag{1}$$ for each x and y. This question is from the 2008 ...
1
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5answers
56 views

Continuously Compounded Interest

What exactly does it mean? By continuously compounded it makes me think it is almost like multiplied as time goes on. Could someone also explain what the constant e is and how it originated? Also how ...
-1
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2answers
31 views

In how many days can they do all the work together?

$A$ and $B$ can do a work in $10$ days. $B$ and $C$ in $15$ days. $C$ and $A$ in $30$ days. In how many days can they do it all working together?
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2answers
48 views

Problem in time and work problems

Asghar can do a job in 60 days. Both Asghar and Babar can do the same job in 20 day working together. How many days will it take Babar to do the job alone? The solution is 30 days. Is there a formula ...
2
votes
2answers
51 views

Evaluation of $\mathop{\displaystyle \sum_{r=1}^{n}r\cdot (r-1)\cdot \binom{n}{r} = }$

Evaluation of $\mathop{\displaystyle \sum_{r=1}^{n}r\cdot (r-1)\cdot \binom{n}{r} = }$ $\bf{My\; Try::}$ Given $$\displaystyle \sum_{r=1}^{n}r\cdot (r-1)\cdot \binom{n}{r}\;,$$ Now Using the formula ...
1
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1answer
92 views

Find Solution to an infinite Nested Radicals

How do I find the solution to the following: $$ \sqrt{ 7 - \sqrt{\frac{7}{2} + \sqrt{\frac{7}{4} - \sqrt{\frac{7}{16} + \sqrt{\frac{7}{256} - \ldots}}}}}$$ I first tried looking for a pattern for the ...
5
votes
1answer
60 views

Prove that: $ (a<b<c) \implies (a<\frac{a+b+c}{3}<c) $

Prove that: $$ (a<b<c) \implies (a<\frac{a+b+c}{3}<c) $$ I'm having problem proving these implications (I don't know how they're called in English). Can you tell me what I have to read to ...
0
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1answer
45 views

proof: $a,b,c \in \mathbb{R}, b > a, c > 0 $, $\Rightarrow$ $bc > ac$

i have to prove for $a,b,c \in \mathbb{R}, b > a, c > 0 $, that $bc > ac$ For this i have two different solutions: solution 1) $bc > ac$ $bcc^{-1} > acc^{-1}$ $b > a$ $b - a > 0 ...
0
votes
1answer
37 views

If $f,g:\Bbb R\to\Bbb R$ are distinct linear functions which map $[-1,1]$ onto $[0,2]$ and $h:\Bbb R\setminus\{-1,0,1\}\to\Bbb R$ defined by $h=f/g$

Given two distinct linear functions $f$ and $g$ defined on $\mathbb R$ such that they map $[-1,1]$ onto $[0,2]$ and $h:\mathbb R\backslash \{-1,0,1\} \to \mathbb R$ defined by ...
1
vote
3answers
87 views

Prove $(cd^{-1})^{-1} = c^{-1}d$

I'm working my way through Michael Spivak's Calculus. There's something I don't quite get about proving in general: I have to prove that: 1) $(cd^{-1})^{-1} = c^{-1}d$ 2) $(cd^{-1})^{-1} \cdot ...
3
votes
1answer
59 views

Sum of digits of 2-digit number is 9. If we switch places of digits, we obtain the number whose ratio to the first number is 8:5

Sum of digits of 2-digit number is 9. The ratio of the number to the number with the digits switched is 8:5. What is the number? My try: We have number $10x+y$ Sum of digits: $x+y=9 \implies x=9-y$ ...
2
votes
1answer
51 views

Domain of a Piecewise Function

I've got a piecewise function defined as : $$f(x)=\begin{cases} |2x-1| & x<1\\ x^2-1 & 1 \le x <2\\ \lfloor 3x \rfloor & x \in [2,3) \end{cases} $$ I am trying to find the domain ...
4
votes
1answer
78 views

Solving Equations Containing Floor Functions

Recently I have been struggling with a problem involving the floor function. The problem is: $$ \lfloor x+5 \rfloor = 3\lfloor x\rfloor-1 $$ I have had a similar question to this however it only ...
1
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3answers
50 views

How can I prove that if $\lim_{n \to \infty}s_n=s$ then $|s_n-s|< \epsilon$ is equivalent to $s-\epsilon <s_n <s+ \epsilon$

My professor casually mentioned this in class and told us to prove it if we weren't convinced, however, I cannot find how to prove it.
0
votes
0answers
16 views

The maximum value for b, when a tangent line to $f(x)=x^{4}-6x^{2}$ at a point $(a, f(a))$ intersects the y-axis at a point $(0,b)$?

How to calculate the maximum value for b, when a tangent line to $f(x)=x^{4}-6x^{2}$ at a point $(a, f(a))$ intersects the y-axis at a point $(0,b)$? How to approach solving this problem?
0
votes
2answers
104 views

The probability that each delegate sits next to at least one delegate from another country

Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate ...
0
votes
1answer
44 views

Mind refresher on a few simple algebra-geometry problems

I feel silly for asking this, but I've completely forgotten some steps on how to do a few of these simple algebra/geometry problems. 1) Simplify $\sqrt{18x}-4\sqrt{x^3}$. I rearranged this to ...
0
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2answers
44 views

What does a duplicate-triplicate-etc ratio mean?

So, if I have three numbers such that : $\dfrac ab = \dfrac bc$ Then we have $\dfrac ac$ which is a duplicate ratio of of $\dfrac ab$ If we have $4$ numbers such that : $\dfrac ab=\dfrac bc=\dfrac ...
4
votes
5answers
293 views

How to prove $3^\pi>\pi^3$ using algebra or geometry?

It's a question of a some time ago test, I've found a way to solve the problem using calculus, but always I've thought that exist a solution with algebra and geometry. Thank you for your time.
10
votes
6answers
906 views

If $3x^2 -2x+7=0$ then $(x-\frac{1}{3})^2 =$?

If $3x^2-2x+7=0$ then $$\left(x-\frac{1}{3}\right)^2 =\text{?}$$ I am so confused. It is a self taught algebra book. The answer is $\large -\frac{20}{9}$ but I don't know how it was derived. ...
1
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4answers
44 views

Factoring Quadratic equation

I am trying to factor $9x^2-6x+1$ after finding the roots, I am using the following formula $a(x-x_1)(x-x_2)$ in this case there is just one root ($\frac{1}{3}$) How do I know that the answer is ...
1
vote
2answers
43 views

Prove that the line $CQ$ passes through a fixed a point

Given $A(3,0)$ and $B(6,0)$ are $2$ fixed points and $P(x,y)$ is a variable point. $AP$ and $BP$ meet the y axis at $C$ and $D$ respectively. The line $OP$, $O$ being the origin intersects the line ...
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3answers
63 views

Math trinom help [closed]

$9x^2-9$ its like $(3x+3) (3x-3)$ what about $9x^2-35$ ?
9
votes
1answer
123 views

Coeff. of $x^{97}$ in $f(x) = (x-1)\cdot (x-2)\cdot (x-3)\cdot (x-4)\cdot …(x-100)$

If $f(x) = (x-1)\cdot (x-2)\cdot (x-3)\cdot (x-4)\cdot ........(x-100)\;,$ Then Coefficient of $x^{99}$ and Coefficient of $x^{98}$ and Coefficient of $x^{97}$ in $f(x).$ $\bf{My\; try::}$ ...
17
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3answers
548 views

When are algebraic expressions equivalent?

This question arose when I was going to determine the domain for $f \circ f(x)$. Let $f(x) = \dfrac{1-x}{1+x}$. $f \circ f(x) = x, \quad$ But the domain is not $\mathbb{R}$ because $f(x)$ is undefined ...
0
votes
2answers
79 views

Solve for $x$: $2^x=4x$

Given that $x$ is a positive integer. By using methods of trial and error as well as plotting two lines: $y=2^x$, $y=4x$ on a graph and find their intersection point, we can easily solve for $x$ which ...
4
votes
2answers
257 views

for which positive integer $m$ does $(ab)^{2015} = (a^2 + b^2)^m$ have positive integer solutions [closed]

For which positive integers $m$ does the equation $(ab)^{2015} = (a^2 + b^2)^m$ Have positive integer solution ?
4
votes
4answers
164 views

How to expand $(x_1 + x_2 + x_3 + x_4 + x_5 +\cdots+x_n)^{2}$

How to expand $(x_1 + x_2 + x_3 + x_4 + x_5 +\cdots+x_n)^{2}$. Is their any general formula for this? Thanks
1
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2answers
45 views

How to solve the equation $x^3+y^3=0$ for real numbers $x$ and $y$?

I'm finding stationary points of the function $f(x,y)=2(x-y)^2-x^4-y^4$, but stuck in the equation $x^3+y^3=0$ while solving the equations $f_x=0$ and $f_y=0$. Please help me. Thanks in advance.
0
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1answer
33 views

Angle of view based on height and distance to a determined object

I'm trying to determine what angle of view is needed for a photo shoot so that I can determine which super telephoto lens to rent. I'm photographing an object thats 2,600 meters across from an ...
4
votes
2answers
104 views

Let $a,b,c>0$ so that $a+b+c=1$…

Let $a,b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove that $$\frac{a}{b}+\frac{b}{a}+\frac{b}{c}+\frac{c}{b}+\frac{c}{a}+\frac{a}{c}+6\geq 2\sqrt{2}\left ( ...
5
votes
4answers
157 views

Why are there only 2 solutions for $x^n=1$?

(where $n>0$) I have been taught that an equation with the highest power $n$ will always have $n$ solutions. This does not appear to be the case with: $$x^n=1 \implies x=\pm1$$ Where $n$ is even, ...
1
vote
1answer
35 views

Finding values of $a$ with which a simple system has exactly 2 solutions

The problem is: Find such values of $a$ with which the system will have exactly two solutions I understand the solution provided at the Resuhege.ru website (problem no. 484630): First ...
0
votes
1answer
20 views

Converting word problems with speed into algebra

'A rower travels upstream at $6$ km per hour and back to the starting place at $10$ km per hour. The total journey takes $48$ minutes. How far upstream did the rower go?' I'm struggling turning the ...
2
votes
4answers
124 views

Why does basic algebra provide one value for $x$ when there should be two?

I have the equation $x^2=x$. If I divide $x$ from both sides I get $x=1$. Yet clearly $x$ can also equal $0$. What step in this process is wrong? It seems to me that there's only one step. And ...
3
votes
5answers
57 views

Quadratics question

To solve $-3x^2 +2x +1=0$, I'd normally break the middle term and then factorise. But I was wondering if there was a way to skip the factorising step? The factors I'd use in place of the middle term ...