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
0answers
7 views

Find the population

Every year, the emigration rate from country A to B is 𝛼 (0 < 𝛼 < 1), whereas the emigration rate from country B to A is 𝛽 (0 < 𝛽 < 1). Note that the fluctuation in population of both ...
1
vote
3answers
24 views

Finding the roots of a different Quadratic equation from the roots of a Given Quadratic equation

The Question: If $\alpha$ and $\beta$ are the roots of the equation $ax^2+bx+c=0$... Then find the roots of the equation $ax^2-bx(x-1)+c(x-1)^2=0$ My Attempt: The new equation can be ...
2
votes
3answers
29 views

Square roots equations

I had to solve this problem: $$\sqrt{x} + \sqrt{x-36} = 2$$ So I rearranged the equation this way: $$\sqrt{x-36} = 2 - \sqrt{x}$$ Then I squared both sides to get: $$x-36 = 4 - 4\sqrt{x} + x$$ Then I ...
2
votes
1answer
17 views

Find the inverse of the function given:

$$f(x)= \frac{5x}{(x − 2)}$$ My work: $$y=\frac{5x}{(x-2)}$$ $$x=\frac{5y}{(y-2)}$$ $$x(y-2)=5y$$ $$xy-2x=5y$$ $$\frac{xy-2x}{5}=y$$ $$f(x)=\frac{(xy-2x)}{5}$$ Any help is appreciated.
0
votes
2answers
17 views

If the i-th, j-th, and k-th terms in an AP are in a GP with ratio r, find $r$ in terms of $i, j$, and $k$

If the i-th, j-th, and k-th terms in an arithmetic progression are in a geometric progression with ratio r, find r in terms of i, j, and k. This is my result: (1) if $ik \ne j^2$ then ...
2
votes
3answers
41 views

According to Stewart Calculus Early Transcendentals 5th Edition on page 140, in example 5, how does he simplify this problem?

In Stewart's Calculus: Early Transcendentals 5th Edition on page 140, in example 5, how does $$\lim\limits_{x \to \infty} \frac{\dfrac{1}{x}}{\dfrac{\sqrt{x^2 + 1} + x}{x}}$$ simplify to ...
0
votes
3answers
28 views

What is the unknown angle?

So first off I started with the pythagorean theorem to find the missing leg of the triangle. \begin{align*} 5^2 + b^2 ={}& 8^2 \\ 25 + b^2 ={}& 64 \\ 64 - 25 ={}& 39 \\ \text{missing ...
1
vote
2answers
32 views

Representation in the complex plane.

Determine the set of representations in the complex plane for which: (a) $\frac{z-1}{z+1}$ is a real number; (b) $\frac{z-1}{z+1}$ is a pure imaginary number.
-2
votes
1answer
44 views

Polynomials and geometric progression? [on hold]

Dividing $P(x)=a_5x^5 + 2x^4 + a_4x^3 + 8x^2 - 32x + a_3$ by $x-1$, we get $Q(x)=b_4x^4 + b_3x^3 + b_2x^2 + b_1x + b_0$, and $6$ as the remainder. Knowing $(b_4,b_3,b_2,b_1)$ is a geometric ...
1
vote
1answer
31 views

Question on Arithmetic and Geometric Progression

Problem: $S_1, S_2, S_3, \ldots ,S_n$ are the sum of $n$ terms of $n$ GPs whose first term is 1 in each case. However, the common ratios $r$ are $1,2, 3, \ldots,n$ respectively. Prove that ...
0
votes
1answer
53 views

How do I prove this nice inequality $x+3\sqrt[3]{xy^2}\geq4\sqrt{xy} $?

Let $x,y\geq0$. Prove that: $$ x+3\sqrt[3]{xy^2}\geq4\sqrt{xy} $$ Note: It's seems easy but when I tried to show it I went to complicated formula.
3
votes
4answers
122 views

Arithmetic and Geometric Progression Question 1

Problem: The second, third and sixth terms of an arithmetic progression are consecutive terms of a geometric progression. Find the common ratio of the geometric progression. My attempt: I ...
4
votes
2answers
73 views

Show that in any triangle, we have $\frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right),$

Show that in any triangle, we have $$\frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right),$$ where $R$ is the circumradius of the triangle. Here is my work: ...
2
votes
1answer
55 views

Find 4 numbers which create a ratio

Find four numbers which create ratio if its known that sum of first and last is equal to 14, sum of middle two is equal to 11 and sum of squares of all numbers is equal to 221 I got only that sum of ...
2
votes
3answers
62 views

How can $f(x)=x^4$ have a global minimum at $x=0$ but $f''(0)=0$?

$f(x)=x^4$ has a global minimum in $\Bbb R$ at the point $x=0$, but $f''(0)=0$. This case confuses me. For every $0\neq x\in I$, $f(x)>f(0)$. So how can it be that $f''(0)=0$, following $f'(x)$ ...
2
votes
3answers
54 views

Is there a simple way to define the $n$-th roots of the unity?

Is there a simple way to calculate the $n$-th roots of the unity? I gotta solve the equation $$\frac{z+1}{z-1}=\sqrt[n]{1}.$$
1
vote
3answers
217 views

A confusion in a calculation with complex numbers

Consider the followings: $$ 1+e^{ix}+e^{2ix}+e^{3ix}= \dfrac{1-e^{4ix}}{1-e^{ix}} $$ Then, we take absolute square to the both sides $$ |1+e^{ix}+e^{2ix}+e^{3ix}|^{2}= \dfrac{1-\cos4x}{1-\cos x} $$ ...
0
votes
1answer
24 views

How do I solve a polynomial equation with a square root variant?

Here is the equation: http://i.stack.imgur.com/UARLE.jpg I notice that the quadratic formula has been used to get the final answer. But something does not seem right because this the equation which I ...
1
vote
2answers
41 views

Fraction with numbers [on hold]

In an audience of people watching a film, 1/10 are under 16 years old, 3/5 are between 16 and 40 years old and the rest are over 40 years old. What fraction are over 40 years old? How to get the ...
0
votes
0answers
26 views

Properties of exponentiation proof

I'm trying to prove the following: "Let $x, y$ be non-zero rational numbers, and let $n,m$ be integers. Then we have $x^n x^m = x^{n+m}$." I've managed to prove by induction the case $n,m \geq 0$ ...
4
votes
1answer
39 views

Roots of $p(x)=\prod_{i=1}^{2n}(x-d_i)+k^2, \ \ \ \ n\in\mathbb N,\ k\in\mathbb R$

Let $$p(x)=\prod_{i=1}^{2n}(x-d_i)+k^2, \ \ \ \ n\in\mathbb N,\ k\in\mathbb R$$ where $d_i>0$ for all $i=1,\dots,2n$. Can I infer that $$p(x)=0$$ has only roots with positive real part?
0
votes
0answers
26 views

Prove base times altitude is a constant without resorting to area.

The area of a triangle is one-half base times altitude. This implies that, for $\bigtriangleup ABC$, $ah_A = bh_B=ch_C$, where $h_A$ is the length of the altitude dropped from point A to side BC, etc. ...
-5
votes
1answer
42 views

Select group of real numbers $x$ [on hold]

Select group of real numbers $x$, satisfy the inequality $$\frac{4x^2}{(1-\sqrt{2x+1})^2}< 2x+9$$ help guys!!
0
votes
2answers
21 views

Confused about proof of division

I thought I was familiar with the regular euclidian algorithm, but I am having trouble understanding a step in this proof from my notes, I am looking for any clarification. $\mathbf{Thereom:}$ Let ...
0
votes
1answer
24 views

Looking to understand proposition related to the fundamental theorem of algebra

I am having some problem understanding exactly what the following proposition is saying. Also, is this result have a common name? How important it is, etc. It is $\mathbf{Proposition:}$ Let ...
0
votes
2answers
49 views

Find six triples of positive integers $(a, b, c)$ such that in $ \frac{9}{a} + \frac{a}{b} + \frac{b}{9} = c$.

Solve for $a, b$ and $c$ in the following equation such that Find six triples of positive integers (a, b, c) such that $$ \frac{9}{a} + \frac{a}{b} + \frac{b}{9} = c$$ I have tried various ...
-2
votes
1answer
40 views

Solve for b and d

Solve for b and d in the following equation. A triangle with sides $(a, a, b)$ has the same area and the same perimeter as a triangle with sides $(c, c, d)$ where $a, b, c$ and $d$ are positive ...
2
votes
0answers
52 views

What is the maximum amount of solutions to $f(x+1)f(x)= ax^2+bx+c$.

$f(x)$ is going to be in the form $mx+h$ thus, $(mx+m+h)(mx+h) = ax^2+bx+c$. With basic algebra $m= \pm \sqrt{a}$. Also $(m+h)(h)=c$. I would guess that because $(m+h)h=c$ has two solutions max if $m$ ...
2
votes
5answers
67 views

Can't determine if given relation is equivalence relation

Definition of relation ~ $(a,b)$ ~ $(c,d)$ $\iff$ $bc^2=da^2$, where $(a,b),(c,d)$ are from $\mathbb{R}\times\mathbb{R}$ and $(a,b),(c,d)$ are different from $(0,0)$ First of all, I wonder if ...
2
votes
4answers
117 views

What is the value of the expression: $(1+\frac 12)(1+\frac 13)(1+\frac 14)…(1+\frac {1}{2004})(1+\frac {1}{2005})$?

What is the value of the expression: $(1+\frac 12)(1+\frac 13)(1+\frac 14)...(1+\frac {1}{2004})(1+\frac {1}{2005})$? This question appeared on the UKMT senior maths challenge 2005, and I can't find ...
0
votes
0answers
61 views

How to calculate the Minimum of a set of Complex numbers? [duplicate]

Suppose you have 5 complex numbers $$2+4i,\ 6-3i,\ -9-7i,\ -12+23i,\ 3+4i.$$ How do you calculate the Minimum? And does it even make sense? If so, what would be a real world example? Thanks, Shane
0
votes
6answers
193 views

If, $x+y=1, x^2+y^2=2$ Find $x^7+y^7=??$

If, $x+y=1, x^2+y^2=2$ Find $$x^7+y^7=??$$ any help guys please?
1
vote
2answers
20 views

Solving for x using two derivatives and algebra.

There are two things I don't understand about the following: " Set these derivatives equal to each other and solve the resulting equation. $2\sqrt3\cos(x) = 2\sin(x)$ $= \sqrt3 = \tan(x)$ (since ...
4
votes
1answer
22 views

Arrangement of any number of objects from $n$ objects

Prove that the total number of arrangements of objects by taking any number of objects from $n$ different objects is $\lfloor e \times n! - 1 \rfloor$, where $e$ is the natural base. I tried it ...
1
vote
2answers
31 views

Positive roots of polynomial $q(x)=p(x)+k^2$

Let $p(x)$ a polynomial of degree $n\in\mathbb N$ such that $$p(x)=0$$ has exactly $n$ real and positive solutions. Is it true that polynomial $q(x)=p(x)+k^2$, for $k\in\mathbb R$ has only positive ...
1
vote
2answers
60 views

Solving an equation that contains a logarithm

I have the follwing equation: $$y=\frac 1 4x^2 -\frac 1 2 \ln{x}$$ How can $x$ be expressed in terms of $y$?
0
votes
2answers
23 views

Resultant Temperature

Ok im not totally sure if this problem can be solved without the theories of physics; but here goes: With three different unknown quantities x,y and z of the same kind of liquid of temperatures 9, ...
3
votes
3answers
54 views

Find the smallest possible value for: $a+b$ [on hold]

If $a,b$ are positive integers with $a, b > 1$, and $$\sqrt{a\sqrt{a\sqrt{a}}}=b,$$ find the smallest possible value for $a+b$.
0
votes
0answers
51 views

IF $x^y=y^x$, Find $x,y$ [duplicate]

If $$x^y=y^x \in\mathbb {R}$$ Find $x,y$. Any help guys?
0
votes
1answer
44 views

I'm trying to solve for a stopping time given a distance. Think I have the answer.

Trying to work with grouping variables and eliminating the exponent. Please help by explaining how you come to a different answer. The equation is $870t=16t^2$ My logic is to divide $t$ from both ...
-4
votes
1answer
23 views

How long will it take two clocks to show the same time once again? [on hold]

There are two analog wall clocks on a wall. On 1st January 2000 daytime, John sees the watches through a mirror placed on the opposite wall showing 10:30 A.M. and 1:30 P.M. respectively. The first ...
0
votes
2answers
132 views

Joining two graphs

Suppose I have $f_1(x)=x$ And i restrict its domain as $\color{blue}{(-\infty,0]}$ using $g_1(x)=\dfrac{x}{\frac{1}{2\left(x-0\right)}\left(x-0-\left|x-0\right|\right)}$ Resulting in : Now, ...
3
votes
2answers
54 views

Plot of $y=x+0\sqrt{-x}$ (and WolframAlpha vs Desmos)

To plot the graph of $y=x+0\sqrt{-x}$ : Do we have to first find out the domain of $y$ which is $y \in ( -\infty,0 ]$ ? $\color{blue}{\text{[Case 1]}}$ (that's what I do) Or do we solve the ...
4
votes
1answer
103 views

Trigonometric ratio of multiple and sub multiple angles

Given that $a$ lies in 1st quadrant and $$ \sin a +\cos a +\operatorname{cosec} a+\sec a+\tan a+\cot a=7$$ then we have to prove that $\sin(2a)$ is a root of $$x^2-44x-36.$$ I have tried to break all ...
0
votes
1answer
10 views

Constructing exponential function using a table of outputs

I have been given the exponential function $g(x)=ar^{x}$. I have also been given the table $(x=4,g(x)=\frac{256}{3})$, and $(x=5,g(x)=\frac{1024}{9})$.... Now as far as I understand you can take ...
4
votes
1answer
76 views

Find the remainder when the sum is divided by $1000$

Find $S \pmod{1000}$ given: $$S = \sum_{n=0}^{2015} n! + n^3 - n^2 + n - 1$$ $$S_0 = 0! + 0 - 0 + 0 -1 = 0$$ $$S_1 = 1! + 1 - 1 + 1 - 1 = 1$$ $$S_2 = 2! + 8 - 4 + 2 - 1 = 7$$ This isn't ...
0
votes
2answers
18 views

Use algebra to decide the rectangle's area

I understand that with the usage of variables, I can use algebra to come up with the right area for the blue rectangle. So I let all the different sides be different variables. Now I know that I ...
0
votes
3answers
66 views

Problem with simplifying $\frac{(3+h)^2-9}{(3+h)-3}$ [on hold]

I need help simplifying $$ {(3+h)^2-9\over (3+h)-3}. $$ The answer is $6+h$. I keep getting $h$.
0
votes
0answers
38 views

Considering bank-interest and inflation rates to calculate remaining money in the account

Peter has A [35,000₤] in bank and banks gives B [350₤] per month as interest; he immediately puts C [100₤] back to the to account and spend the rest of it R [250₤] till next months. Every month, ...
0
votes
2answers
35 views

Finding the parameter a [on hold]

The ratio of the roots of the equation $x^2 +ax + a+2=0$ is $2$ Find the values of parameter $a$. I don't understand what the question means .