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
votes
1answer
33 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
vote
1answer
24 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 ...
-6
votes
2answers
59 views

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

Please help solving $4^{x-1}+2^{x-2} = 68$.
0
votes
1answer
26 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
62 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
46 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 ...
3
votes
1answer
85 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
vote
5answers
48 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
votes
2answers
27 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?
0
votes
2answers
25 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
48 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
vote
1answer
45 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
46 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
votes
1answer
44 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
32 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
79 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
40 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
35 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 ...
3
votes
1answer
46 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
vote
3answers
49 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
13 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
89 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
29 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
votes
2answers
24 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 ...
3
votes
5answers
209 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.
7
votes
6answers
508 views

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

If $3x^2 -2x+7=0$ then $(x-\frac{1}{3})^2 =$ ? I'm so confused. It's a self taught algebra book. The answer is $-\frac{20}{9}$ but I don't know how it was derived. Please explain. Thanks for ...
1
vote
4answers
38 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
35 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 ...
-4
votes
3answers
47 views

Math trinom help [closed]

$9x^2-9$ its like $(3x+3) (3x-3)$ what about $9x^2-35$ ?
8
votes
1answer
108 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
votes
3answers
481 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
69 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
122 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 ?
3
votes
4answers
81 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
vote
2answers
42 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
votes
1answer
17 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
89 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
143 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
34 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
14 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
72 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
53 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 ...
1
vote
1answer
20 views

Create a set of system of linear equations to answer the following.

A factory is currently running at $85\%$ of its original capacity, and management is considering upgrading the equipment. The upgrade will take $6$ months, during which time the factory will not ...
5
votes
3answers
50 views

Line for set of three-dimensional vectors

If there is a set for 3D vectors $v$ where $ v \times \begin{pmatrix} -1 \\ 1 \\ 4 \end{pmatrix} = \begin{pmatrix} 5 \\ -27 \\ 8 \end{pmatrix}$ is a line, what is this line's equation? I'm not sure ...
2
votes
1answer
47 views

Let $a^n = a^{n - 1} + a^{n -2}$. Show that for any $A, B$, $F(n) = Aa^n + Bb^n$ satisfies Fibonacci recurrence relation.

$$\begin{align*} F(n) &= Aa^n + Bb^n\\ &= A(a^{n-1}+a^{n-2}) + B(b^{n-1}+b^{n-2}) \\ &= Aa^{n -1} + Aa^{n-2} + Bb^{n -1} + Bb^{n-2}\\ &= a^{n -1} (A + A^{a-1}) + b^{n - 2} (B + bB) ...
0
votes
1answer
34 views

Find the number of sets satisfying the conditions

Let $ N$ be the number of ordered pairs of nonempty sets $ \mathcal{A}$ and $ \mathcal{B}$ that have the following properties: • $ \mathcal{A} \cup \mathcal{B} = ...
1
vote
1answer
28 views

Finding a Recurrence Relation.

This is from AMC 2015 . For each positive integer n, let S(n) be the number of sequences of length n consisting solely of the letters A and B, with no more than three As in a row and no more than ...
2
votes
1answer
38 views

How to find solutions for this nonlinear equation?

I want to find an analytical solution $x$ as a function of parameters $(e,u,r,t)\in\mathbb{R}^4$ that satisfies the following condition: ...
1
vote
1answer
36 views

Find circumradius of $\Delta DEC$

$A(0,0),B(4,0)$ and $C(5,-2\sqrt 6)$ are the vertices of $\Delta ABC$. Incircle of the triangle touches side $AC$ and $BC$ at $D$ and $E$ respectively. Find the circumradius of the triangle $DEC$. Is ...
3
votes
3answers
47 views

Number of Non - Decreasing functions?

Let A={1,2,3.....10} & B={1,2,3....20}. We have to find the number of non decreasing functions from A-->B. What I tried :No. Of non decreasing functions = (Total functions) - (Number of ...