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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2
votes
2answers
31 views

Limit $\lim_{\left(x,y\right) \rightarrow \left(0,0\right)} \frac{x^3+y^3}{\sin x^2+y^2}$

Find the limit of: $$\lim_{\left(x,y\right) \rightarrow \left(0,0\right)} \frac{x^3+y^3}{\left(\sin x^2 \right)+y^2}$$ How to find this limit? What is the most straightforward method?
0
votes
4answers
58 views

Highschool Algebra: $n^2 = 18n$?

I'm beginning to get into maths outside of school and at the moment I'm refreshing myself on the basics which explains why this question appears to be so simple. I formulated this equation to find ...
2
votes
2answers
237 views

Seemingly Simple Equation Question “Verify my solution please!”

"A container is $1/8$ full of water. After $10$ cups of water are added, the container is $3/4$ full. What is the volume of the container, in cups?" Ok, I wrote out an equation: $\frac{1}{8}V + 10C = ...
1
vote
2answers
28 views

Prove that from the equalities, $\frac{x(y+z-x)}{\log x}=\frac{y(x+z-y)}{\log y}=\frac{z(y+x-z)}{\log z}$ follows $x^yy^x=y^zz^y=z^xx^z$.

Problem : Prove that from the equalities, $$\frac{x(y+z-x)}{\log x}=\frac{y(x+z-y)}{\log y}=\frac{z(y+x-z)}{\log z}$$ follows $$x^yy^x=y^zz^y=z^xx^z$$. My approach : $$\frac{x(y+z-x)}{\log ...
0
votes
1answer
24 views

Can't figure out this basic algebra

Been a while since I did math but I'm trying to understand how they got the final equation in this step: http://i.imgur.com/Y09bqwT.png When I solve for P I get this: $$ P(t) = ...
-3
votes
1answer
25 views

A plant can manufacture 50 golf clubs per day at a total daily cost of $ \$5423$ and $70$ golf clubs per day for a total cost of $ \$6,923$.

Assuming that daily cost and production are linearly related, find the total daily cost, $C$, of producing $X$ gold clubs Interpret the slope and $Y$-intercept of the cost equation. I have no idea ...
2
votes
2answers
26 views

Determine polynomial whose roots are a linear combination of roots of another polynomial

Let $\alpha_1, \alpha_2, \alpha_3$ be the roots of the polynomial $p(x)=x^3+5x^2+7x+11$. Find a polynomial whose roots are $\frac{\alpha_1+\alpha_2}{2}, \frac{\alpha_2+\alpha_3}{2}, ...
0
votes
2answers
16 views

Use the difference quotient to compute a formula in terms of h

Using the difference quotient: $\frac{f(x+h) - f(x)}{h}$, I need to compute a formula in terms of $h$, given $f(x)$ and $x$, ensuring that the $h$ in the denominator gets cancelled out. Given an ...
0
votes
2answers
33 views

Solving an equation with $\sin(x)$ in the exponent: $2^{\sin(x)} \cdot \cos(x) + 1 = 1$

Hi I need help with a trig problem: I have $2^{\sin(x)} \cdot \cos(x) + 1$, and I need this to equal $1$ between $x = -3$ and $3$. I keep going in circles with substitution, etc. Any help would be ...
-3
votes
1answer
21 views

How do you solve these for Intersection Points [on hold]

How do you get the intersection points between these algebraically? $$\sqrt{4-y} = (y-2)^2+2$$
-4
votes
2answers
18 views

math problem two possible values [on hold]

Find the two possible values of x that make the expression true. (2x – 6) (x + 5) = 0
1
vote
3answers
75 views

Showing for any real number $\lfloor a\rfloor+1>a$

This seem a simple proposition For any real number a $\lfloor a\rfloor+1>a$ For any example $\lfloor 2.9\rfloor=2$ $\lfloor 3.1\rfloor=3$ $\lfloor 4\rfloor=4$ I think this is obvious. Because ...
0
votes
2answers
52 views

Function that transforms the interval $[a,b]$ into $[0,1]$ [on hold]

Could someone please give me an example of function that translates the interval $[a,b]$ into $[0,1]$ I tried $\frac{x-a}{p(x)}$ and after that $x(x+b-a-1)+a$.
-3
votes
0answers
32 views

I have 70 gallons of water at 660 ppm. I need to bring the ppms up using 30 gallons of water, what does my ppm for the 30 gallons need to be? [on hold]

I have 70 gallons of water at 660 ppm, I need to use 30 gallons to bring the ppms up, what ppm does my 30 gallons need to be at ? Trying to bring the ppms in the 70 gallons up to 770?
2
votes
3answers
24 views

Considering Units When manipulating system of Equations?

I few days ago I solved a problem on a website called brilliant.org, I can not seem to find the problem there anymore but I still remember it: Q: You go to a candy store to buy m&ms and ...
0
votes
2answers
32 views

Find the solution set of the equation $5.(\frac{1}{25})^{\sin^2x}+4.5^{\cos2x}=25^{\frac{\sin2x}{2}}$

Problem : Find the solution set of the equation $5.(\frac{1}{25})^{\sin^2x}+4.5^{\cos2x}=25^{\frac{\sin2x}{2}}$ where $x \in [0,2\pi]$ My approach : ...
1
vote
0answers
16 views

$n$-tuples of points of $\mathbb{C}$, identification.

Fix $n \in \mathbb{N}$. Forgive me if this is a very silly question, but how can I see that the set of unordered $n$-tuples of points of $\mathbb{C}$ can be naturally identified with $\mathbb{C}^n$?
2
votes
4answers
84 views

Find a Polynomial in x-1/x

Given that $x^n - (1/x^n)$ is expressible as a polynomial in $x - (1/x)$ with real coefficients only if $n$ is an odd positive integer, find $P(z)$ so that $P(x-(1/x)) = x^5 - (1/x)^5.$ To start, I ...
3
votes
0answers
60 views

Evaluate $\int \dfrac{1}{\sqrt{x-1}+\sqrt{x}+\sqrt{x+1}} \ \mathrm{d}x$ [duplicate]

Evaluate $$\int \dfrac{1}{\sqrt{x-1}+\sqrt{x}+\sqrt{x+1}} \ \mathrm{d}x$$ I tried rationalizing the denominator by twice multiplying, but it didn't do any good. I also tried trig ...
3
votes
1answer
68 views

Prove that $s(n-1)s(n)s(n+1)$ is always an even number

Let $n$ be a natural number, and let $s(n)$ denote the sum of all positive divisors of $n$. Show that for any $n>1$ the product $s(n-1)s(n)s(n+1)$ is always an even number. I calculated the sum of ...
1
vote
1answer
35 views

ASTC: Finding exact values of trigonometric functions

Our teacher showed us this really dodgy way of finding exact values by drawing up the 4 ASTC (all stations to central diagram) quadrants and making a right angle to the x axis. So how would I do a ...
1
vote
1answer
44 views

About a matrix identity.

In a document named as "The Matrix Cook-Book" I saw two expressions of which I do not get any clue how they are derived. For $n = 3:$ $\det(I + A) = 1 + \det(A) + Tr(A) + 1/2\ Tr(A)^2 − 1/2\ ...
-2
votes
3answers
41 views

write an expression [on hold]

A word processor determines the width of the body of text on a page. The page is 11 inches wide and has two equal size margins of x inches on each side of the text. Write a formula that gives the ...
3
votes
3answers
80 views

Why does $e^{-x}$ approach $0$ as $x$ gets large? [on hold]

Why is it that $$\lim_{x \to -\infty} e^x = 0?$$
2
votes
1answer
269 views

Probability or Set

I'm really good at probability, but this time I seems like I'm not. My friends asked me a very tricky question, and I want to see if there's anyone who can find out the answer. Here's the ...
2
votes
6answers
57 views

Limit of $\lim_{x \rightarrow 0} \frac{\sin xy^2}{x}$

Limit of $$\lim_{x \rightarrow 0} \frac{\sin xy^2}{x}$$ I know (thanks to wolfram) it is equal to $y^2$, but i do not know how to show that.
1
vote
2answers
62 views

trying to solve $\sqrt{\cos(x)-2\cos(2x)}+\sqrt{2}\cos(2x)=0$

The equation is $$\sqrt{\cos(x)-2\cos(2x)}+\sqrt{2}\cos(2x)=0$$ The system is $$ \begin{cases} \cos(x)-2\cos(2x)=2\cos^2(2x) \\ -\sqrt{2}\cos(2x)\ge 0 \iff \cos(2x)\le 0 \end{cases} $$ The ...
0
votes
2answers
39 views

Find the limit of $\frac{x+y+\sin xy}{x^2+y^2+\sin^2 (xy)}$

Find the limit of: $$\lim_{(x,y)\rightarrow(+\infty, +\infty)}\frac{x+y+\sin xy}{x^2+y^2+\sin^2 (xy)}$$ I think the solution could be: $$\frac{x+y+\sin xy}{x^2+y^2+\sin^2 (xy)} \le \frac{x+y+\sin ...
1
vote
1answer
48 views

Algebra Problem: Division

Can someone help me with a problem involving the expression $$\frac{(2x^3-3x^2+b)}{(4-x^2)}?$$ The question is to find which values $b$ can be to simplify the expression, but I do not know how to ...
0
votes
8answers
96 views

$x(x^2-2)=0$, The answers are $x = 0, \sqrt{2}$, how do I get there? [on hold]

$$x(x^2-2)=0$$ The answers are $x=\sqrt{2}, 0$ how do I get there?
3
votes
2answers
27 views

Minimum of $f(x)=\sum_{i=1}^n\frac{a_n}{x-b_n}$ occurs at extreme point?

Let $a_1,\ldots,a_n$ be real numbers and $b_1,\ldots,b_n>1$. Define $$f(x)=\sum_{i=1}^n\frac{a_i}{x-b_i}.$$ Is it always true that $f(x)\geq\min\{f(0),f(1)\}$ for all $x\in[0,1]$?
0
votes
2answers
27 views

Changing equation to x equals

Im currently stuck on this equation I need to modify to be in terms of x $$y=-x^2+4$$ I got something like this which looks wrong $$x = -\sqrt{y+4}$$ First you would subtract the 4 from both ...
1
vote
3answers
33 views

Distance/Speed word problem

A train of length 300m can cross a pole in 8 seconds. How long will it take to cross a platform of length 600m. I can't seem to appreciate the very beginning. Crossing the pole implies that the time ...
1
vote
2answers
32 views

An 11-gon with complex numbers

Let $A_1 A_2 \dotsb A_{11}$ be a regular $11$-gon inscribed in a circle of radius $2$. Let $P$ be a point, such that the distance from $P$ to the center of the circle is $3$. Find $[PA_1^2 + PA_2^2 ...
0
votes
2answers
40 views

Geometric progression (compound interest)

"A man, who started work in 1990, planned an investment for his retirement in 2030 in the following way. On the first day of each year, from 1990 to 2029 inclusive, he is to place £100 in an ...
1
vote
2answers
83 views

Trying to solve $\sqrt{7-4\sqrt2 \sin x}=2\cos(x)-\sqrt2 \tan(x)$

The equation is $$\sqrt{7-4\sqrt2 \sin x}=2\cos(x)-\sqrt2 \tan(x)$$ We get the system $$ \begin{cases} 7-4\sqrt 2 \sin(x)=4\cos^2(x)-2\sqrt2\cos(x)\tan(x)+2\tan^2(x) \\ 2\cos(x)-\sqrt2 \tan(x)\ge 0 ...
-1
votes
1answer
30 views

Months for the amount paid to be equal? [on hold]

If I take my pension now I can get $\$2,000$ a month. If I wait $12$ months I can get $\$2,500$. How many months will it take for the total amount paid to me be equal?
0
votes
2answers
35 views

Help with a progress bar algorithm for a website

I have a progress bar in a website that needs to be filled based on the number $50$. So at $50$ it will be $100\%$ full. The problem is that it starts at about $20\%$ then follows this pattern : ...
7
votes
1answer
120 views
+50

Find all pair of cubic equations

Find all pair of cubic equations $x^3+ax^2+bx+c=0$ and $x^3+bx^2+ax+c=0$, where $a,b$ are positive integers and $c$ not equal to $0$ is an integer, such that both the equations have three integer ...
4
votes
5answers
80 views

Does the limit $\lim\limits_{x\to0}\left(\frac{1}{x\tan^{-1}x}-\frac{1}{x^2}\right)$ exist?

Does the limit: $$\lim\limits_{x\to0}\frac{1}{x\tan^{-1}x}-\frac{1}{x^2}$$ exist?
1
vote
1answer
32 views

Solving two equations with 2 variables

I am wondering if this equations can be solved by "a" and "b": b = 1 + 0.31*a a = c1 - c2/b c1 and c2 are constants, but change depending on some initial assumptions. One example of their ...
0
votes
3answers
50 views

Simplifying Cube Roots Containing a Square Root

I was doing a problem today, and arrived at the (correct) answer of $x^3 = 16000\sqrt2$ Obviously I want to simplify this further. My text book jumps straight to $x = 20\sqrt2$ with no explanation. ...
2
votes
3answers
60 views

Give the equations that are a tangent to the parabola $y = x^2 + 5x + 6$ and pass through $(1,1)$

I have been given the question: Give the equations that are a tangent to the parabola: $y = x^2 + 5x + 6$ and pass through the point $(1,1)$ I have tried two different methods for solving this. ...
2
votes
4answers
71 views

Is there an integer solution to $x^2+1978=y^2$

Is there an integer solution to $x^2+1978=y^2$? Don't know really how to approach this. Thanks
0
votes
5answers
62 views

Are these two expression equal?

My friend insisted that $(-1)^{(-n)}$ is equivalent to $(-1)^n$ for any number of $n$. A quick check in the Wolfram Alpha show ...
3
votes
4answers
62 views

Trying to solve the trig equation $\sqrt{3+4\cos^2(x)}=\frac{\sin(x)}{\sqrt 3}+3\cos(x)$

The equation is $$\sqrt{3+4\cos^2(x)}=\frac{\sin(x)}{\sqrt 3}+3\cos(x)$$ My solution goes like this $$ \begin{cases} 3+4\cos^2(x)=\frac{\sin^2(x)}{3}+\frac{6}{\sqrt 3}\sin(x)\cos(x)+9\cos^2(x) \\ ...
0
votes
2answers
32 views

Getting two different sets of results for $\sqrt{17+7\sin(2x)}=3\sin(x)+5\cos(x)$

The equation is $$\sqrt{17+7\sin(2x)}=3\sin(x)+5\cos(x)$$ My solution is, first, to define a system: $$ \begin{cases} 17+7\sin(2x)=(3\sin(x)+5\cos(x))^2 \\ 3\sin(x)+5\cos(x)\ge 0 \end{cases} $$ ...
0
votes
2answers
32 views

What is wrong with this formula?

I'm trying to make a formula that converts an ellipse in general form to one in standard. My steps to derive it are as follows: $$ax^2+bx+cy^2+dx+e=0$$ Move e to the other side... ...
2
votes
0answers
33 views

Let $A_1 A_2 \dotsb A_{11}$ be a regular 11-gon inscribed in a circle of radius 2.

Let $A_1 A_2 \dotsb A_{11}$ be a regular 11-gon inscribed in a circle of radius 2. Let $P$ be a point, such that the distance from $P$ to the center of the circle is 3. Find $PA_1^2 + PA_2^2 + \dots ...
-1
votes
0answers
37 views

Equilateral triangle [on hold]

An equilateral triangle is one in which all three sides are of equal length. If two vertices of an equilateral triangle are $(0,\,4)$ and $(0,\,0)$, find the third vertex. How many triangles are ...