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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0
votes
2answers
30 views

Show$\:\frac{1}{\left|x^2+x+1\right|}\:\ge \:\frac{1}{x^2-\left|x\right|-1}$

This is the answer I can come up with. I get the complete opposite of what I'm supposed to get. My mistake is probably in the first part, could anyone help me out? $$\left|x^2+x+1\right|\:\ge ...
-3
votes
3answers
22 views

The lines with equations $y = 5x − 6$ and $10x + cy = 8$ are perpendicular, find c

The lines with equations $y = 5x − 6$ and $10x + cy = 8$ are perpendicular. Find the value of c. Well, I am not sure even where to start
0
votes
1answer
43 views

$(a+\frac{1}{2})^n + (b+ \frac{1}{2})^n$ is an integer for at most finitely many $n$ [duplicate]

Prove that for any positive integers $a,b$ $(a+\frac{1}{2})^n + (b+ \frac{1}{2})^n$ is an integer for at most finite number of integers $n$. Here is what I tried ; I tried to use mathematical ...
-4
votes
1answer
31 views

Algebra,complex numbers home work problem [on hold]

Please I want the solution of this problem : $z= \dfrac{(2-i) \cdot (x+4i)}{3-4i}$ and $|z|=2$ then $X=?$
0
votes
5answers
46 views

Number theory proof [on hold]

$(i)$ Prove that for every natural number $n \geq 2$, one has $(n + 1)|(n^3 + 1)$; $(ii)$ Suppose that $n$ is a natural number exceeding $1$. Prove that $(n^2-1)|(n^3+1)$ if and only if $n = 2$.
3
votes
0answers
27 views

Rationality and triangles

Consider a triangle with angles $\alpha, 5\alpha, 180-6\alpha$. What is the minimum perimeter of that triangle, if it has integer sides and $5\alpha<90$?. Let's call tha sides that face each ...
0
votes
1answer
41 views

what is going on here?

Suppose we have a function $f(x), D:( -\infty,0)\cup (0,\infty)$ and for which $$f'(x) = \frac{x^3-1}{x^3} $$ Apparently there is only one point of extremum here, $x=1$, however upon reviewing the ...
0
votes
1answer
31 views

Polynomials, prove exercise question about question

There is a polynomial P with integer coefficients and with pairwise different integers $a,b,c$ . Prove that it is not possible for $P(a) = b$, $P(b)=c$, $P(c) = a$ First off I don't understand ...
1
vote
1answer
22 views

Finding an Expression for the Difference of Roots of the Quadratic Equation

Let the equation $ax^2+bx+c=0$ have the roots $\alpha$ and $\beta$, then what is $\alpha-\beta$ in terms of $a$, $b$, and $c$? Well, we may write $$(\alpha-\beta)^2=(\alpha+\beta)^2 -4\alpha \beta$$ ...
0
votes
1answer
13 views

solve $k(k-1) \geq \ln2*2m$ for k

My Question is related to the birthday problem. Starting at $e^{-\frac{k(k-1)}{2m}} \leq 0.5$ i used $ln(x)$ on both sides and multiplied by $-2m$ to get $k(k-1) \geq \ln2*2m$ According to my ...
-2
votes
0answers
16 views

Find the slope of the secant line given a point [on hold]

The point P(1,0) lies on the curve y= sin(10π/x) . (a) If Q is the point (x, sin(10π/x), find the slope of the secant line PQ (correct to four decimal places) for x=2, 1.5, 1.4, 1.3, 1.2, 1.1, 0.5, ...
1
vote
0answers
28 views

How can I solve this para Paradox? [duplicate]

How can I solve this para Paradox? $ -1={(-1)}^{1/2} {(-1)}^{1/2}={[(-1)(-1)]}^{1/2}=1$
-7
votes
4answers
72 views

Can $9+10$ equal $21$? [duplicate]

I just saw a video with the following: $9+10=21$ because $$9 = 3*3$$ $$10 = 5*2$$ if $$5*3=15$$ and $$2*3=6$$ $$15+6=21$$ does that not prove that $10+9=21$?
1
vote
1answer
25 views

Which functions are power functions? [on hold]

I have 6 functions and want to know which ones are power functions and which ones aren't. I know that the power function has the form $f(x)=Kx^p$, where $K$ and $p$ are constants. $f(x)= \pi*x^4$ ...
-1
votes
2answers
18 views

Find closed form for a sequence using the Fibonacci and Lucas number sequences. [on hold]

Define $A_n$ as follows: $$\begin{align*} A_0 &= 6 \\ A_1 &= 5 \\ A_n &= A_{n - 1} + A_{n - 2} \; \textrm{for} \; n \geq 2. \end{align*}$$ There is a unique ordered pair $(c,d)$ such that ...
-10
votes
0answers
55 views

Solving equation please help [on hold]

Solve for $x$ $$-6(x-7)=-2(4+3x) $$
4
votes
2answers
96 views

How do I simplify this expression about factorization?

I am trying to simplify this $$\frac{9x^2 - x^4} {x^2 - 6x +9}$$ The solution is $$\frac{-x^2(x +3)}{x-3} = \frac{-x^3 - 3x^2}{x-3} $$ I have done $$\frac{x^2(9-x^2)}{(x-3)(x-3)} = ...
1
vote
1answer
14 views

Creating a weighted score

I have an audit where there are six criteria, each can be scored Excellent (E), Satisfactory (S), Needs improvement (N) or Unsatisfactory (U). I know that if someone scores Excellent in all six areas ...
0
votes
2answers
50 views

Matrix invertible iff det(matrix)$\neq 0$?

When we want to find the inverse of the matrix $$\begin{bmatrix}a & b \\ c & d\end{bmatrix}$$ we're searching for a matrix $$\begin{bmatrix}x & y \\ z & w\end{bmatrix}$$ such ...
2
votes
1answer
33 views

Differents between $lnx^2$ and $ln(x^2)$ Find derivative

I have this problem Find derivative for $lnx^2$. It seems that $lnx^2 \neq ln(x^2)$ since the derivative are differents using Wolfram Alpha. I don't understand how to calculate the derivative for ...
2
votes
7answers
621 views

Working Out Easy Equations

does anyone know how to do this equation? I know it's easy but I can't work out what the question means. When I expanded the first equation: $(y+4)-(y-3)$ $y^2 -3y +4y - 12$ $y^2-1y-12$ Not ...
0
votes
1answer
39 views

Getting ready for Calculus?

So I wanted to start a Masters program but they require that I have Calculus III. I want to take that course at the university, but I need to be ready for it. As I look at Khan Academy and do some ...
2
votes
1answer
29 views

Calculating $\sum_{k=0}^{n-1}\frac{1}{a+bk^2}$.

I want to calculate the following summation: $$\sum_{k=0}^{n-1}\frac{1}{a+bk^2}$$ Any hint how I can calculate this? Is there any kind of closed form for this summation?
2
votes
1answer
30 views

Algebraic values of the sine function

First question: For which angles $x$ is $\sin(x)$ a real number that can be expressed using only integers, addition, subtraction, multiplication, division and the extraction of $n$th roots? (With ...
2
votes
1answer
30 views

Inequality $(a+b)^2 + (a+b+4c)^2\ge \frac{kabc}{a+b+c}$ for $a,b,c \in\mathbb{R}$

Find biggest constans k such that $(a+b)^2 + (a+b+4c)^2\ge \frac{kabc}{a+b+c}$ is true for any $a,b,c \in\mathbb{R}$ Could you check up my solution? I'm not sure it's ok - $(a+b)^2 + (a+b+4c)^2 \ge ...
2
votes
1answer
48 views

Inequality $x^4+y^4+(x^2+1)(y^2+1)\ge x^3(1+y) +y^3(1+x)+x+y$ for $x,y \in\mathbb{R}$

Prove for $x,y \in\mathbb{R}$ that such inequality exists ; $x^4+y^4+(x^2+1)(y^2+1)\ge x^3(1+y) +y^3(1+x)+x+y$ And here is what I realised ; because $(x^2+1)(y^2+1) >=1$ and $x^4+y^4 \ge 0$ ...
-2
votes
4answers
63 views

How to determine the nth term of a sequence, given only the first four terms? [on hold]

I need to calculate the tenth term of the following sequence: $1 \quad 8 \quad 27 \quad 64\quad \ldots$
3
votes
1answer
35 views

Solving A Certain Diophantine Equation

I am stack on finding the solution of the diophantine equation: $d(2^{k+1}-1)-b^2(2^{k+1}-2)=1$. where $k\geq 1$ and $b^2>d$ for $b$ an odd composite integer. Is there a solution to this ...
-4
votes
1answer
34 views

Linear-algebra problem [on hold]

How do you solve this equation? $$ -7x-28=7+6(1-8x)$$
0
votes
1answer
46 views

How can the modulus of something be less than zero?

I've been asked to prove that for $\epsilon>0$, $$| a-x| < \epsilon \iff a-\epsilon<x<a+\epsilon,$$ and as a hint to consider both $| x-a|>0$ and $| a-x|<0$. I used the fact that ...
2
votes
2answers
56 views

How many positive integers less than $2013$ are divisible by none of $2, 3, 4 ,5$?

How many positive integers less than $2013$ are divisible by none of $2, 3, 4 ,5$? This was an olympiad question. I thought of writing a number $x \le 2012$ in the form: $x = 2^{a}3^{b}4^{c}5^{d} = ...
0
votes
2answers
14 views

FV (Future Value) of annual payments

My client will receive $\$881$ now and twice that amount in a year. He will get 3 times $\$881$ in 2 years . 4 x $\$881$ in 3 years etc. for as long as he lives. Assuming he lives 20 years and each ...
0
votes
2answers
46 views

sum of exponentials to non-integer power

I have the expression \begin{equation} (e^{at}+e^{bt}+e^{ct})^{v} \end{equation} for some a,b,c which isn't important. I'd like to take a limit $t\rightarrow \infty,v\rightarrow 0,vt=\text{constant}$. ...
1
vote
2answers
42 views

how old are Inee and Imee now? [on hold]

Inee and Imee are twins. Their mother is 28 years older than they are and 4 times as old as the sum of their ages. How old are Inee and Imee now?
1
vote
1answer
88 views

Number of real roots of $2 \cos\left(\frac{x^2+x}{6}\right)=2^x+2^{-x}$

Find the number of real roots of $ \cos \,\left(\dfrac{x^2+x}{6}\right)= \dfrac{2^x+2^{-x}}{2}$ 1) 0 2) 1 3) 2 4) None of these My guess is to approach it in graphical way. But equation seems ...
0
votes
1answer
20 views

Simplify $\frac{\sum_{i = 1}^{n}x_{i}}{n} - \frac{n - \sum_{i = 1}^{n}x_{i}}{1 - \theta} = 0$ to show that $\theta = \bar{x}$

Simplify $\frac{\sum_{i = 1}^{n}x_{i}}{n} - \frac{n - \sum_{i = 1}^{n}x_{i}}{1 - \theta} = 0$ to show that $\theta = \bar{x}$ $\frac{\sum_{i = 1}^{n}x_{i}}{n} = \frac{n - \sum_{i = 1}^{n}x_{i}}{1 - ...
-1
votes
1answer
27 views

Simplify the function

I am having problems solving this, any help would be appreciated. Find $f(x+h)-f(x)$ and simplify if $f(x)=2+3-x^3$ Thanks in advance.
0
votes
3answers
22 views

Need help in clarifying relation of square root and logarithm to do a correct substitution

This might be so basic and obvious, but I am stuck on how to do substitution that involves logarithm and square root. If we have $$\lfloor\sqrt{n}\rfloor$$ and we do the following substitution ...
0
votes
0answers
4 views

Order of Dilated horizontally and translated horizontally

I have a parent function $f(x) = x^2$, and $g(x) = (6[x-2]))^2$ is a transformation from $f(x)$. The question is: $g(x)$ is from $f(x)$ by Dilated horizontally by a factor of 1/6, then translated ...
-1
votes
2answers
27 views

how many jelly beans did each girl have at first?

Martha and Mary had $375$ jelly beans in all. After Mary ate $24$ jelly beans and Martha ate $\frac 17$ of her jelly beans, they each had the same number of jelly beans left. How many jelly beans did ...
0
votes
0answers
28 views

Quention about the historical definition of determinant

$$ax+by = k_1\\cx + dy = k_2$$ If I want to solve for $y$ in the first equation: $$by = k_1 - ax\implies y = \frac{k_1-ax}{b}$$ Then substitute $y$ in the second equation: $$cx + d\frac{k_1-ax}{b} ...
1
vote
3answers
82 views

Let $p^3+q^3=4$ and $pq=2/3$ . Find $p+q$.

Let $p^3+q^3=4$ and $pq=\frac{2}{3}$ . Find $p+q$. A graphing calculator can find values of $p$ and $q$ numerically. As one can see from the graph below, the two solutions are approximately ...
-5
votes
4answers
52 views

24 hours before Wednesday [on hold]

I have a procedure scheduled for 11 a.m. on Wednesday. I can't take certain medications for 24 hours, so what time should I be able to take my last dose?
0
votes
4answers
41 views

Finding maximum of a function with unknown constants

I have a function in the form: $$y = \frac{ax}{b + \frac{x^2}{c} + x}$$ Supposedly, the maximum of this function is equal to $\sqrt{bc}$. I've tried substituting in $\sqrt{bc}$ for $x$, but I don't ...
0
votes
4answers
63 views

How many solutions has this third degree equation?

how many solutions has this equation: $$ {x}^{3}+4\,{x}^{2}-1=0 $$ i tried ruffini so far and it is not working, now i'm stuck and no idea of how to aproach this.
0
votes
0answers
26 views

Integer solutions to an equation

Let $x,y,z$ be positive integers and $S$ be the set of all the solutions to the equation $x^y+y^z=z^x$. Is $S$ finite or infinite? Lots of thanks for any help in advance.
0
votes
0answers
11 views

Find a cyclic rational function such that…

I'm looking for a function of the form $\frac{f(a,b,c)}{f(b,c,a)}$ (or close to this form, e.g. $\frac{(a+b)^2}{b^2+bc+c^2}$) which is roughly equal to $\frac{b^3-a^2-b^2-a^3-ab^2}{b^2c+a^2b+b^3}$ (I ...
3
votes
2answers
69 views

Prove, inequality ,positive numbers

$$\frac{a}{e+a+b}+\frac{b}{a+b+c}+\frac{c}{b+c+d}+\frac{d}{c+d+e}+\frac{e}{d+e+a}<2$$ Prove that for positive numbers $a,b,c,d,e$ there is such inequality
-1
votes
3answers
23 views

How can we make this expression small? [on hold]

How can we make the following expression small: $$(bx-ay)^2+(cx-az)^2+(cy-bz)^2+(ay-bx)^2+(az-cx)^2+(bz-cy)^2$$, where $a,b,c,x,y,z$ are nonnegative reals? Note: I'm not looking for an exact answer, ...
2
votes
2answers
49 views

divide 6 people in group of 2 in same size

Exercise: divide 6 people in group of 2 in same size. My solution: The exercise tells us to calculate the combination without repetition. If I start by calculating the number of ways to select how ...