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.

learn more… | top users | synonyms (2)

0
votes
0answers
6 views

If $x \in \{1,2,3, \cdots, 9\}$ and $f_n(x) =xxxx\cdots x($ n digits) then find the value of $f_n^2(3)+f_n(2)$

If $x \in \{1,2,3, \cdots, 9\}$ and $f_n(x) =xxxx\cdots x($ n digits) then find the value of $f_n^2(3)+f_n(2)$ If x =1, then $f_n(1) = ?$ Please suggest how to expand such series. Thanks
0
votes
1answer
22 views

Beautiful logical minimum construction

On a circle after equal intervals 25 points are located. On every point is a policeman. All policemen are numbered (from 1 to 25) in some way. Now they have to move to some other points through this ...
0
votes
2answers
31 views

Prove by induction that $r_0 + r_1a + r_2a^2 + \cdots + r_{n−1}a^{n−1} < a^n$.

Let $a$ be a natural number greater than $1$. Prove that for all integers $r_0 , r_1 , \cdots , r_{n−1}$ with $0 ≤ r_j < a$, we have: $$ r_0 + r_1a + r_2a^2 + \cdots + r_{n−1}a^{n−1} < a^n ...
1
vote
1answer
39 views

Beauiful wordly combinatorics.

Show that the number of m-letter words using letters T O W N(maybe not using some of them) in which the number of Ts and Os are equal, is equal to the number of 2m-letter words only using Ts and Os ...
4
votes
1answer
55 views

Product of numbers that remains invariant repeatedly when adding one to all of them

Eugene wrote a 100 numbers on a board. He added 1 to each number and the product didn't change. He did the same thing k times, each time the product didn't change. What is the maximum k? I guess the ...
0
votes
1answer
22 views

Find $f(n)$ such that $f(m + n) + f(m - n) = 2f(m) + 2f(n)$ for all integers $m$ and $n$ and $f(4) = 16$

The question is in the title itself- The function $f(n)$ takes the integers to the real numbers such that $f(m + n) + f(m - n) = 2f(m) + 2f(n)$ for all integers $m$ and $n$ and $f(4) = 16$. Find ...
2
votes
1answer
14 views

Finding an upper bound for this expression

Given two non-negative numbers $a$ and $b$, I'm trying to found an upper bound for $-\sqrt{a} + \sqrt{a+b}$ But I'd like the bound not to be dependable on any square roots and have only one term. ...
2
votes
1answer
41 views

Multiplying and adding fractions

Don't be angry with me. Just comment to delete if you think this is really a bad question. Why multiplying fractions is equal to multiply the tops, multiply the bottoms? $$\frac{a}{b}\times ...
0
votes
2answers
26 views

Number of values that satisfy $2\sin ^2(x) - 3 = 3 \cos (x), \: 90^{\circ} < x < 270^{\circ} $

Graphing this function is difficult as many overlaps exist and finding a viewing window is hard. What's a good algebraic method to solve this problem?
-3
votes
2answers
50 views

High school Math Team problems. [on hold]

In triangle $\triangle ABC$, $AB=5,BC=6$ and $AC=7$. The circle with diameter $AB$ intersects $BC$ at $D$, ($D \neq B$). Compute $BD$.
-4
votes
0answers
25 views

Sin and Cos help please!! [on hold]

Find the Sin and Cos of an angle whose terminal points pass through (4,-8)
0
votes
2answers
20 views

Finding the domain of this trigonometric function

How can I find the domain of this function? $$f(x)=\frac{x\sin(x)+\cos(x)}{1-\cos(x)} + \frac{|x|-2}{x^2-4}$$ I assume we don't want the denominator to be zero, but do we have to combine the ...
1
vote
2answers
21 views

Finding the equation of vertical and horizontal asymptotes

I am having some trouble understanding these two questions. Any help is appreciated. Scanned questions are included at the end. 6) We are given the function $ f(x) =\frac{1 - 2x} {2x^2 - 3x - 2} $ ...
1
vote
2answers
18 views

Square root and principal square root confusion

A few months ago I asked a question about the $\pm$ symbol because I was confused about it... I still carry the same confusion (which really bugs me) but I think the real confusion has to do with the ...
0
votes
3answers
66 views

How to evaluate this $1/n$ infinite sum?

How to evaluate$$\sum ^{\infty}_{n=1} {e}^{-n}$$ without using the easy-formula. We easily notice a pattern. $$\begin{align} S_1 &= e^{-1} \\ S_2 &= e^{-2} + e^{-1} = \frac{1 + e}{e^2} \\ ...
0
votes
1answer
22 views

Maths Problem based on trains

If A train runs at $70$km/h, it reaches its destination late by $12$ minutes. But if it runs at $80$km/h it is late by $3$ minutes. What is the correct time to cover the journey?
0
votes
3answers
37 views

quadratics equation tricky problem

I am confused with this question- if $ax^2+bx+c$ have no real roots then- $1+c/a+b/a$ is-- a. Positive b. Negative c. Zero d. Can.t say I tried attempting it as follows $b^2-4ac<0$ so ...
0
votes
2answers
21 views

How to show that $at^2+bt+c$ can be written as $\displaystyle \begin{equation} a \left( t+\frac{b}{2a}\right) ^2-\frac{1}{4a}(b^2-4ac)\end{equation}$?

I've just expanded $$\displaystyle \begin{equation} a \left( t+\frac{b}{2a}\right) ^2-\frac{1}{4a}(b^2-4ac)\end{equation}\tag{1}$$ to $$at^2+bt+c\tag{2}$$ but I guess that perhaps showing it ...
0
votes
2answers
54 views

Solving $(x-2)^2=\sqrt{x}+2$

How to solve $(x-2)^2=\sqrt{x}+2$ answer is $0.9$ and $4$ But no idea with steps
-2
votes
2answers
52 views

Find the number of natural numbers less than 2014 which are neither squares nor cubes [on hold]

Find the number of positive integers less than 2014 that are neither squares nor cubes.
2
votes
2answers
79 views

$\cos ^2(x)=\frac{1}{2} \cos (2 x)+\frac{1}{2}$

I am wondering how this is rewritten? $$\cos ^2(x)=\frac{1}{2} \cos (2 x)+\frac{1}{2}$$, I think it has something to do with double-angle/half angle? but I am not sure and I do not see the ...
7
votes
1answer
46 views

Newton's way of getting a Taylor expansion

I don't understand how Newton find the Taylor expansion of $\frac{a^2}{b+x}$ by the following method : **This screenshot is from : The method of fluxions and infinite series Any idea ?
0
votes
2answers
34 views

For which values of $m$, $f(x)=mx$ intersect the function $g(x)=\log x$?

For which values of $m$, the function, $f(x)=mx$ intersect the function, $g(x)=\log x$ I suppose that this problems reduce to the next form. Find for which values of m, exist solution for the ...
0
votes
1answer
37 views

Simultaneous function with three variables using subsititution method

Use any substitution method and solve the following equations: $$2x+5y+7z=86$$ $$3x+y+5z=60$$ $$x+4y+3z=54 $$ I used $x+4y+3z=54$ to make $x$ the subject $x=54-4y-3z$.
-2
votes
0answers
21 views

Nursing Home Question [on hold]

A 1-year old 120-bed for-profit facility with a \$2,500,000 mortgage at 7.4% reports a net operating margin before depreciation of \$25,000. The answer is that in today's financial market the owners ...
0
votes
1answer
36 views

Find the range of function $f(x) =\cos(\sin(\ln(\frac{x^2+e}{x^2+1})))+\sin(\cos(\ln(\frac{x^2+e}{x^2+1})))$…

Problem : Find the range of function $f(x) =\cos(\sin(\ln(\frac{x^2+e}{x^2+1})))+\sin(\cos(\ln(\frac{x^2+e}{x^2+1})))$ My approach : maximum value of the function is when denominator term is ...
2
votes
1answer
51 views

Beautiful evaluation inequality

a,b,c>0 are such that $a^2+b^2+c^2=4$ and $4(a^2+2)=(a^2+b+c)^2$. What is the biggest possible value for a+b+c? I tried a lot of stuff like $a^2=4-b^2-c^2$. And i think it's somehow connected to ...
0
votes
0answers
17 views

Real world situation/model using (factored) higher degree polynomial function?

(Form A) $$y=3x^5+23x^4-7x^3+x^2-4x+9$$ (Form B) something in factored form $$y=(x-1)^2(x+2)^5(x-5)^4(x+7)$$ 2 part question: 1) Anyone know of examples where these types of functions arise? 2) Is ...
2
votes
1answer
39 views

simple SAT practice question

I tried to solve this and I am getting 11n/6. Am I seriously doing something really wrong or is this question wrong?
1
vote
5answers
100 views

Is 1^2^3 = $1^{2^3}$ or $(1^2)^3$ [duplicate]

Caret ^ signs can be used to describe the power of numbers. Is $1$^$2$^$3 = 1^{(2^3)}$ or $(1^2)^3$ How do you calculate it? Do you start with $2^3$ and then do $1^8$ or do you start with $1^2$ ...
0
votes
2answers
26 views

Trigonometry: How to determine the Period

I'm still kinda confused with solving the period on the diagram above. Amplitude= $3$ Max = $3$ Min = $-3$ Period = ? $y=a\cos(bx+c)$ Value of $a$ = $3$ Value of $b$ = ? Value of $c$ = ?
6
votes
1answer
80 views

Solve this tough fifth degree equation.

$$x^5+x^4-12x^3-21x^2+x+5=0$$ I think it can be solved by trigonometric ways but how?
1
vote
1answer
36 views

To show $(x+y)^p\leq x^p+y^p$, where $0\leq p\leq1, x>0,y>0$? [duplicate]

How to show that, $(x+y)^p\leq x^p+y^p$, where for $0\leq p\leq 1,x\geq 0, y\geq0?$ Any suggestion how to prove it? Thanks in advance.
1
vote
2answers
68 views

Remainder of $3^7/8$

I read here that the remainder of $\frac{ab}{c}$ is equal to the remainder of $\frac{a}{c}\frac{b}{c}$ implying that the remainder of $\frac{a^b}{c}$ is equal to the remainder of $[\frac{a}{c}]^b$. ...
1
vote
4answers
40 views

quadratic equation: $5x^2 + 9x - 170 = 0$

I have a problem, my textbook says the solution of $5x^2 + 9x - 170 = 0$ is $5$ but the book didn't describe how it solved the equation. How can I solve this?
2
votes
4answers
70 views

Find the sum of an infinite series of Fibonacci numbers divided by doubling numbers. [duplicate]

How would I find the sum of an infinite number of fractions, where there are Fibonacci numbers as the numerators (increasing by one term each time) and numbers (starting at one) which double each time ...
2
votes
2answers
39 views

Is there a way to get the closed form of $x$ considering $x^2 + 3x = \sqrt{x + 2}$, not using calculator/computer?

How would one go about finding the exact answer to $$x^2 + 3x = \sqrt{x + 2}$$ Solving for $x$? Using paper and pencil to plot a graph, I've found the solution lies at $\approx 0.453$, but I am ...
-4
votes
3answers
57 views

Solving $x^2-6=\sqrt{x+6}$ [on hold]

Solving $x^2-6=\sqrt{x+6}$ Thanks for any help
1
vote
0answers
54 views

Solving equations, Math olympiad, using vieta relation?

So the question asks to solve for real valued $a$ such that $b,c,d\in\mathbb{R}$ $$abcd=-1$$ $$(a+c)(b+d)=-1$$ $$ac+bd+a+b+c+d=-1$$ $$ab+cd=ac+a+c$$ So assuming the four numbers are roots of a quartic ...
2
votes
2answers
22 views

Find one set of solutions for the following system:

Find one set of solutions for the following system: \begin{cases} 1+a^2+d^2=3+b^2+e^2=3+c^2+f^2 \\ 1+ab+de=0 \\ ac+df=0 \\ bc+ef=0 \\ \end{cases}
2
votes
2answers
39 views

Probability of 8 or 9 digit sequence colliding in the same place in two 65 digit numbers

I have two numbers: 3032643431333337636238613038343231383364303731376566303037663231 3861663464383131656131653461343961343364303737663565356561653361 36430373 ...
7
votes
1answer
45 views

How do I find a constant for a polynomial so its roots are reflective around a linear function?

How can I find all complex numbers $w$ so that the roots of the following polynomial are reflected around a linear function $f(x)$ $$p(q) = q^2-4q+w = 0$$ If I want to find all the complex numbers ...
-2
votes
0answers
27 views

Explain why every natural number (other than one) is divisible by at least one prime number? [on hold]

Explain why every natural number (other than one) is divisible by at least one prime number? This is through Euclid's idea. A prime number is a natural number with exactly two distinct positive ...
3
votes
4answers
179 views

Solve $\frac{x^2+2xy+y^2}{x^2-y^2} >x+y$

Find the set of integer solutions $(x,y)$ to $$\frac{x^2+2xy+y^2}{x^2-y^2} >x+y$$ I can't seem to multiply both sides by the expression in the denominator. Nor can I simplify and cancel any ...
-1
votes
1answer
13 views

Trapezoids and Bases [on hold]

A trapezoid has bases of length $x$ and $4$. Let $P$ and $H$ be points on opposite legs of the trapezoid. $PH$ is parallel to the bases and divides the trapezoid into two quadrilaterals of the same ...
0
votes
2answers
28 views

Finding relationship between two numbers directly that were changed cumulatively

Bear with me. I'm not sure how to express this question let alone answer it. Here goes... I have a program that can calculate change from a single rate, we'll call 'A'. Known: $$C = A + 400$$ ...
1
vote
1answer
19 views

Third degree polynomial with unknown coefficients $q^3-3aq^2+b^2q+c = 0$

For an equation $q^3-3aq^2+b^2q+c = 0$ we know the roots $c, (a+b), (a-b)$. What is a good place to start with such equations? I've tried setting up a system of equations, but this is supposed to be ...
-1
votes
1answer
59 views

Mathematically prove that a bench which 2 chidlren fit in can't fit 3. [on hold]

You have a bench( Only 2 children can sit on it), 3 children and you have to prove logically that 3 children don't fit on the bench.
1
vote
2answers
23 views

How do I solve for $\delta$ in this question

$316.45 = 100e^{\delta(10)} + 100e^{\delta(5)}$ I don't know why I can't do this. I thought of using $\ln$ but I don't think $\ln(A+B) = \ln(A) + \ln(B)$ or does it?
0
votes
4answers
82 views

How do I solve this one? It's irritating me!

How do I solve this question? I can't think of anything to do! As $(x,y)$ ranges over all pairs of real values, what is the smallest value of: $(2x-3y-4)^{ 2 }+(2x-3y+10)^{ 2 }$