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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3
votes
1answer
47 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
90 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
219 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
516 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
39 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
114 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
486 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
71 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
156 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
82 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
44 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
90 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
146 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
15 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
55 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
32 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
48 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 ...
1
vote
1answer
47 views

Solve the equation: $(9x^2+6x-8)\sqrt{3x+2}+6x+23=27x^2+3\sqrt{10+3x}$

Solve the equation: $(9x^2+6x-8)\sqrt{3x+2}+6x+23=27x^2+3\sqrt{10+3x}$ I used wolframalpha.com and got only solution $x=-\dfrac{1}{3}$. And this is my try: Condition: $x\ge-\dfrac{2}{3}$. ...
3
votes
1answer
37 views

Prove that $\left (\sum_{k=1}^{n}\frac{1+x^{2k}}{1+x^{4k}} \right )\left ( \sum_{k=1}^{n}\frac{1+y^{2k}}{1+y^{4k}} \right )< \frac{1}{(1-x)(1-y)}.$

Let $n$ be a positive integer, and let $x$ and $y$ be positive real numbers such that $x^{n}+y^{n}=1$ Prove that $$\left (\sum_{k=1}^{n}\frac{1+x^{2k}}{1+x^{4k}} \right )\left ( ...
-1
votes
1answer
38 views

Finding values of $a, b$ such that $0\le x^4 +x^3 +ax+b\le (x^2-1)^2$

Given real values of $a, b$ such that for all $x\ge0$, $$0\le x^5+x^3+ax+b\le (x^2-1)^2\ ,$$ find the value of $ab$. What I've done is let $x=1$, thus $$0\le2+a+b\le0$$ this forces $a+b=-2$. let ...
0
votes
2answers
78 views

find the value of $x^{(x^{2})}+x^{(x^{8})}$

If $$x^{(x^{4})}=4$$ Then find the value of $$x^{(x^{2})}+x^{(x^{8})}$$ I did solve this, I want to see more solutions, thanks
1
vote
2answers
35 views

quadratic reduction problem

A train is travelling between two stations that are $100$ km apart at a speed of $v$ km/h. Express the time taken for the journey in terms of $v$. Here I got $\ t=\dfrac{100}{v}$. On the return ...
3
votes
2answers
79 views

Why am I getting two answers for 8th root of continued fraction

Find value of $x$: $x=\sqrt[8]{2207-\frac{1}{2207-\frac{1}{2207-....and\,so\, on}}}$ On solving ,we have $x^8=2207-\frac{1}{x^8}$ $x^8+\frac{1}{x^8}=2207$ $x^4+\frac{1}{x^4}=47$ ...
0
votes
2answers
56 views

How do I isolate/solve for $\theta$ in $\sin (2\theta) = 4 \cos (2\theta)$

Isolate the variable/solve for $\theta$: $$\sin (2\theta) = 4 \cos (2\theta)$$ Like which $\cos$ double angle formula would I use? Because there are three of them. Thanks in advance.
0
votes
3answers
27 views

If the coefficients of the y-terms are equal and the coefficients of the x-terms are equal, the graphs of the two lines will be parallel.

So a student has a claim that for any pair of linear relations, if the coefficients of the y-terms are equal and the coefficients of the x-terms are equal, the graphs of the two lines will be ...
3
votes
1answer
56 views

Maximize the Cyclic sum

Let $x_1,x_2,\dots ,x_6$ be nonnegative real numbers such that $x_1+x_2+x_3+x_4+x_5+x_6=1$, and $x_1x_3x_5+x_2x_4x_6 \geq \frac{1}{540}$. Let $p$ and $q$ be positive relatively prime integers such ...
2
votes
2answers
33 views

find the limit of $ \lim_{(x,y) \rightarrow (0,0)}\frac{2xy^3+x^2y^3}{x^4+2y^4}$

find the limit of $$ \lim_{(x,y) \rightarrow (0,0)}\frac{2xy^3+x^2y^3}{x^4+2y^4}$$ I have absolutely no idea how to proceed with that. I would prefer a solution that would involve use of squeeze ...
0
votes
1answer
37 views

finding the value of x from a complex form of absolute value

How do I find the value of x for this one, $$ |x-3|^{\frac{x+1}{4}} = |x-3|^{\frac{x-2}{3}}$$ I tried equating the exponents when I found out that the base of both sides are equal but I don't think ...
-1
votes
2answers
66 views

Problem involving geometric progression [closed]

Question: The bacteria in a certain culture double every $7.3$ hours. The culture has $7,500$ bacteria at the start. How many bacteria will the culture contain after $3$ hours? Possible Answers: a. ...
1
vote
1answer
53 views

a very basic question on finding the discriminant for $x^2+2(a-3)x-3a-7=0$

Sorry for asking such a basic question. In the following quadratic equation $$x^2+2(a-3)x-3a-7=0$$ by my calculations, $$D=\left(\frac{b}{2}\right)^2-ac=(a-3)^2-1(-3a-7)=a^2-6a+9+3a+7=a^2-3a+16$$ ...
0
votes
3answers
45 views

limit of $\frac{x^3y+xy^2}{x^2+y^2}$

find the limit of $$ \lim_{(x,y) \rightarrow (0,0)}\frac{x^3y+xy^2}{x^2+y^2}$$ no idea how to deal with that. I've tried to use squeeze theorem however this attempt was unsuccessful.
5
votes
2answers
119 views

Calculate $S=3\sqrt{\sqrt[3]{5}-\sqrt[3]{4}}-\sqrt[3]{2}-\sqrt[3]{20}+\sqrt[3]{25}$

Calculate $$S=3\sqrt{\sqrt[3]{5}-\sqrt[3]{4}}-\sqrt[3]{2}-\sqrt[3]{20}+\sqrt[3]{25}$$ $\color{red}{\text{without using calculator}.}$ Please help me, I can't find any solution to sovle it.
2
votes
2answers
23 views

Then the value of $ [f(2)] $ where [.] represents the greatest integer function is?

A differentiable function f is satisfying the relation $$f(x+y) = f(x) + f(y) + 2xy(x+y) - \dfrac{1}{3} $$ $ \forall $ $ x , y $ belongs to $\Re$ and $$lim_{h \to 0} \dfrac{3f(h)-1}{6h} = ...
0
votes
2answers
32 views

If $f(x) = \max\left|2\sin y-x\right|,$ Then $\min.$ value of $f(x)$

If $f(x) = \max\left|2\sin y-x\right|\;,$ Where $y\in \mathbb{R}\;,$ Then $\min.$ value of $f(x)$ $\bf{My\; try}$ We know that $-2 \leq 2\sin y\leq 2$. Now I did not Understand How Can I open ...
3
votes
2answers
33 views

Area of shaded region circle help

Find the area of the shaded region Area of the sector is $240^\circ$ or $\frac{4\pi}{3}$ Next find $\frac{b\cdot h}{2}$ which is $\frac{2\cdot2}{2}$ which is $2$. Then subtract the former ...