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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1answer
22 views

The Triple Number Game (n + n + n)

I work at the Science Museum on weekends, and I sometimes present the following puzzle: $$0\;0 \; 0 = 6$$ $$1 \; 1 \; 1 = 6$$ $$2 \; 2 \; 2 = 6$$ $$3 \;3 \; 3 = 6$$ $$...$$ $$n\;n\;n=6$$ The idea is ...
1
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1answer
20 views

Proving a reduction formula. $\cos^n (2x)$

Establish a reduction formula for $$\int \cos^n (2x)dx$$ My attempt, Let $I_n=\int \cos^n 2x dx$ $=\int \cos^{n-1}2x (\cos 2x dx)$ Let$$u=\cos^{n-1}2x$$ $$du=-2(n-1)\cos^{n-2}2x (\sin 2x)dx$$ ...
2
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2answers
25 views

Fraction Sum Series

This question was asked in (selection) IMO for 8th graders. $1/2 + 1/6 + 1/12+ 1/20 + 1/30 + 1/42 +1/56 + 1/72 + 1/90 + 1/110 +1/132$ I have noticed that it can be written as $1/(1*2) + 1/(2*3) ...
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0answers
43 views

Are polynomial roots special? [on hold]

Most functions have roots, and relations can also have roots (including complex roots) as well. They are where $f(x)=0$ for some $x$. (I'm considering only functions that actually will have roots.) ...
1
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4answers
68 views

Is there any value for $x$ that would make the statement $(x+3)^3 = x^3+3^3$ true?

Is there any value for $x$ that would make the statement $(x+3)^3 = x^3+3^3$ true? I understand that when factored out, you have $(x+3)^3 = x^3+9x^2+27x+27$ as opposed to the other side which is ...
1
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1answer
23 views

Precision of Manual Vector Addition

I learned the fundamentals of vectors and basic (e.g. addition, dot product) vector operations in a Trigonometry course, and they're being reintroduced in the Physics I course I just began. My ...
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1answer
13 views

Generate a formula to calculate affect on continously compound interest if a constant salary is continously added to principle

I'm feeling stupid with a simple algebra question. I'm hoping someone can help me figure out why something I know should be obvious just isn't clicking for me. I won't go into actual details of the ...
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3answers
69 views

If $a$ is a real root of $x^5 − x^3 + x − 2 = 0$, show that $\lfloor a^6 \rfloor = 3$.

If $a$ is a real root of $x^5 − x^3 + x − 2 = 0$, show that $\lfloor a^6 \rfloor = 3$. Obviously since this is a 5th degree polynomial, solving it is not going to be possible (or may be hard). ...
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3answers
31 views

The line $x+\sqrt{3} y-10=0$ makes an angle of $150$° with the positive sense of the $x$-axis. How can this be proven?

I cant figure out how this is correct. I know that $\tan(a)=m$ of a line but I cant figure this out. Could someone show how to prove the line makes an angle of $150$° with the positive $x$-axis? I ...
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2answers
38 views

Solve $x + \frac{ 1 }{y+1/3}=38/3$ in the set of natural numbers

The following equation should have a solution with $x,y$ being natural numbers. I cannot find it. Is there such solution? $$x + \frac{ 1 }{y+1/3}=\frac{38}{3}$$
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4answers
19 views

Evaluating simple summation

can someone help with this summation. Seems simple, but... I have tried several options but cannot see the rule. $\displaystyle 1-a+a^2-a^3+...a^{2008}-a^{2009}+\frac{a^{2010}}{1+a} {\text{ when}}\ ...
1
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0answers
29 views

reduction formula for $\int \tan^n (2x)dx$

Establish a reduction formula for $$\int \tan^n (2x)dx$$ My attempt, Let $I_{n}=\int \tan^n (2x)dx$ $=\int \tan^2 (2x) \tan^{n-2} (2x)dx$ $=\int (\sec^2 (2x)-1)\tan^{n-2}(2x)dx$ $=\int ...
0
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0answers
23 views

how to scale a value from 0-1 to 0-5

I have these facts: the user (X) rates the item (I) by (4/5) the item (II) is 0.5 similar to item (I), 0.5 means 50% (the scale is from 0 to 1) then I can say (according to my business model) that ...
0
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3answers
15 views

How can I find $y$ coordinate of a straight line at a specific $x$ value

Lets say I have a straight line between $p_1=(-2, -0.5)$ and $ p_2=(0.25, 0.5)$. How can I find the value of $y$ when $x=-1$? I have tried to solve this the whole day without finding an answer, I ...
2
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2answers
53 views

How to solve without involving hyperbolic function.

How to solve this integral without involving hyperbolic functions? $$\int \frac{1}{4-5\sin^2 x}dx$$ The answer is $\frac{1}{4}(\ln (\sin x+2 \cos x)-\ln(2\cos x-\sin x))+c$
3
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4answers
72 views

Show that $8x^4 −16x^3 +16x^2 −8x+k = 0$ has at least one non-real root for all real $k$. Find the sum of the non-real roots

Show that $8x^4 −16x^3 +16x^2 −8x+k = 0$ has at least one non-real root for all real $k$. Find the sum of the non-real roots. Since this polynomial looks so symmetric, I think factoring it might ...
6
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1answer
62 views

Basic question $|x^2| < 9$

I have a rather basic question. Let's assume that $|x^2| < 9$, where $x\in \mathbb{R}$. Then everyone knows that $x \in$ (-3,3). However, I have trouble arriving at the answer based on basic ...
1
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2answers
56 views

Integer value of the given radical: $\sqrt{2+\sqrt{5}-\sqrt{6-3\sqrt{5}+\sqrt{14-6\sqrt{5}}}}$ [on hold]

What is the value of $$\sqrt{2+\sqrt{5}-\sqrt{6-3\sqrt{5}+\sqrt{14-6\sqrt{5}}}}$$ I don't know how to simplify it?
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2answers
52 views

Where did my simplification go wrong? Sum and difference formula simplification

I'm struggling with the following: We are to use the sum and difference formulas to find the exact value of the expression. The problem is simplification has been tough. As a last resort I decided to ...
0
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1answer
21 views

Is there a unique solution to this upstream/downstream canoe rowing proposition?

A man jumped into his canoe and paddled upstream for one mile. After this, he continued for another fifteen minutes. Having arrived at his destination, he then turned around and paddled downstream, ...
2
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5answers
43 views

How to write absolute value as a “true” function

Here's the basic absolute value ... what? \begin{align*} |x| = \left\{ \begin{array}{r@{\quad \mathrm{if} \quad}l} x & x > 0, \\ 0 & x = 0, \\ \!\! -x & x < 0. \end{array} ...
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2answers
25 views

Factor Theorem (Finding values of a and b)

Question: The polynomial $p(x) = 2x^3 - ax^2 + bx + 48$ has $(x-4)$ as a repeated factor, find the values of $a$ and $b$. What I have attempted if $x-4$ is a factor then $x = 4$ is a ...
0
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2answers
37 views

solving system of equations involving imaginary numbers

What are the values of $a,b,c$ given the system of equations given below: $a+b+ab=i$ $b+c+bc=2i$ $c+a+ac=3i$
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1answer
58 views

What is my special quadratic?

Start with $f(x)=x^2+bx+c$. Then, attempt to solve for $x$ in $f(x)=x$. It is easily found that $x=\frac{1-b\pm\sqrt{(b-1)^2-4c}}{2}$. Then, start again with $f(x)=x$ and apply the function $f$ to ...
3
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0answers
17 views

Is this a valid way for performing polynomial division?

While attempting to divide a quartic by a quadratic factor to find the other factors of the given quartic, I can't help feeling I "invented" a way of dividing polynomials. Suppose you have a quartic ...
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0answers
25 views

How do I get these values? [on hold]

I want to know how to get these answers: $A(12836.3)=42227.7$ $A=3.28971$ (phase) $\theta+46.7364=0$ $\theta=-46.7364$ from this equation. I am confuse is there a formula or a trig. identity? ...
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0answers
31 views

Find all real $x$ such that $1990[x] +1989[-x]=1$ (where $[x]$ is the floor function for $x$).

Find all real $x$ such that $1990[x] +1989[-x]=1$ (where $[x]$ is the floor function for $x$). My effort Rearranging our equation we have : \begin{array}{c} 1990[x]+1989[-x]&=1 \\ ...
1
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1answer
30 views

Struggling with BIDMAS.

This question came up an I'm not sure about it: You need to simplify leaving the answer in standard form: $\dfrac{2((3-3^2)^2)}{3+\sqrt{4^2-7}}$ I struggle to work it out myself. When I used a ...
6
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4answers
110 views

Solve equation $\frac{1}{x}+\frac{1}{y}=\frac{2}{101}$ in naturals

My try was $$\frac{1}{x}+\frac{1}{y}=\frac{x+y}{xy}=\frac{2}{101}\\x+y=2k,xy=101k\\x=2k-y\\y(2k-y)=101k\\2ky-y^2=101k\\y^2-2ky+101k=0\\y=k+\sqrt{k^2-101k}\\x=k-\sqrt{k^2-101k}$$ Now $\sqrt{k^2-101k}$ ...
2
votes
1answer
57 views

Why do you need absolute value when taking $\sqrt{\cos^2(x)}$

$$\sqrt{\cos^2(x)} = |\cos(x)|$$ Is this on the right track? If you have an underlying $\cos(x)$ that is negative, and then you square it, you will now have $\cos^2{x}$, which is positive. But, if ...
4
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2answers
47 views

Is there an identity that says $|\sqrt {a^2+x^2} - \sqrt {a^2+y^2}| \leq |\sqrt {x^2} - \sqrt {y^2}|$?

Is there an identity that says $|\sqrt {a^2+x^2} - \sqrt {a^2+y^2}| \leq |\sqrt {x^2} - \sqrt {y^2}|$? Because of the nature of the square root function, its derivative monotonically decreases. so ...
0
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2answers
37 views

What is my mistake

Spot my mistake: $$\frac{\left(\text{P}_1+\text{P}_2+\dots+\text{P}_n\right)-\left(\text{Z}_1+\text{Z}_2+\dots+\text{Z}_n\right)}{n-m}\le-\ln(50)$$ ...
0
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2answers
24 views

Simplifying Expression Factorial Expression

I'm confused as how I'm meant to simplify this:$$\frac{(n-2)!}{(n-2-r)!}$$ I have other factorial questions where the variable isn't present in the top factorial like the question above and I'm ...
7
votes
5answers
68 views

Proving that $\cos(\frac{\arctan(\frac{11}{2})}{3}) = \frac{2}{\sqrt{5}}$

I am trying to solve the cubic equation $x^3-15x-4=0$ using Cardano's formula. I already know that the solutions are $x=4$, $x= \sqrt{3}-2$ and $x= -\sqrt{3}-2$ and that using the formula in this ...
1
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2answers
38 views

$N + 2N + 3N +\dotsb$ series?

I've been trying to figure this out but no luck yet. What is the general formula for the sum $N + 2N + 3N + 4N +\dotsb$? Could it be equal to something like $$N + 2N +3N + ..... + xN = ...
1
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0answers
22 views

Difference Equations and Discrete Integral and Derivative

So I'm trying to learn difference equations, and the book that I'm using defines the following: The discrete derivative of a function $a_n$ of the integers is defined as: $$ D a_n = a_{n+1} - a_n $$ ...
0
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4answers
76 views

Is this argument valid for a proof?

Please kindly forgive me if my question is too naive, i'm just a prospective undergraduate who is simply and deeply fascinated by the world of numbers. My question is: Suppose we want to prove that ...
0
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2answers
52 views

Need help solving an equation: $f(x)=\frac{1}{x}+\ln{x}-2=0$ [on hold]

How would I solve this equation: $$f(x)=\frac{1}{x}+\ln{x}-2=0$$ What I noticed was that $\frac{d}{dx}\ln{x}=\frac{1}{x}$. Don't know how this would help.
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2answers
54 views

Solving algebraically a cubic inequality

Is there any way to solve algebraically the following inequality: $x-x^{3}+x^{2}-5 \geq 0$. I know the answer is $x \leq -1.594$ because I have plotted the function but I can't figure out how to do ...
3
votes
3answers
73 views

What's the equation for a rectircle? (Perfect rounded-corner rectangle without stretching on only one dim)

The equation for a rounded square seems to be: $x^4 + y^4 = 1$ You can make the radii smaller by increasing (over the even integers) the exponents in the equation. Here's a picture: Wolfram Alpha ...
2
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0answers
34 views

A subset of easily solved 4th degree polynomials

I've found (maybe, maybe not, but it's not on this Wikipedia or this Wikipedia) that there is a subset of easily solved quartic polynomials of the form ...
0
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4answers
57 views

number of real solution of $\sin x+2\sin 2x = 3+3\sin 3x\;\forall x\in \left[0,\pi\right]$

Number of real solution of $\sin x+2\sin 2x = 3+3\sin 3x\;\forall x\in \left[0,\pi\right]$ My try: I have tried graphing. Using the graphs of $$y=\sin x+2\sin 2x \;\;\; \text{and} \;\;\; ...
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3answers
36 views

Show that $x^3y^3(x^3+y^3) \leq 2$

If $x,y$ are positive reals such that $x+y = 2$ show that $x^3y^3(x^3+y^3) \leq 2$. I thought about working backwards on this one. We can try to prove that $x^3y^3(x^3+y^3) \leq 2$ is true by ...
1
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1answer
19 views

Distance between a point $X$ and a line $2x-y = 1$

I am asked to state a condition for a point $X$ such that the point $X$ is a distance of 10 units from the line $2x-y = 1$, and then find the Cartesian equation of the set of all points $X$ that are a ...
3
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3answers
86 views

Solve the equation $7t+[2t] =52 $ ,where $[x]$ denotes the floor function for $x$.

Solve the equation $7t+\left\lfloor 2t\right\rfloor =52 $. My effort Using the fact that for any number $x$ we have that $x=\left\lfloor x\right\rfloor+\{x\}$ (where $\{x\}$ is the fractional ...
0
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2answers
13 views

Comparing the asymptotic growth of certain functions.

Consider some nonconstant functions $f(x)$ and $g(x)$, and suppose $\lim_ {x \to a} f(x) > \lim_{x \to a} g(x)$ for some nonzero $a$. Can we somehow conclude that $f(x) - g(x)$ is of constant sign ...
-3
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0answers
24 views

mixing of three type of teas [on hold]

Three tea, whose prices per kg are respectively 15dollars,25dollars and $30, are to be taken two at a time and mixed in the same proportion so that the resulting mixtures are of equal value. How may ...
-2
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2answers
116 views

What does it mean to say “again” or “finally” in math? [on hold]

According to math rules if we say again, does it mean we are saying repeat previous step? For example: You have $10$ coins. Add $10$ more coins. Add $2$ coins Again $2$. Again $2$. Add $2$. And ...
1
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2answers
66 views

If $ \frac{x}{b-c} = \frac{y}{c-a} = \frac{z}{a-b}$, Prove that $x+y+z$ $=0$

Question If $$ \frac{x}{b-c} = \frac{y}{c-a} = \frac{z}{a-b} $$ Prove that $x+y+z$ $=0$ I've attmempted this question by cross multiplying so that $$ x(c-a)(a-b) = y(b-c)(a-b) = ...
0
votes
1answer
44 views

Finding a sixth degree polynomial that goes through 8 points

For a summative math research assignment, I will have to find a sixth degree polynomial that would ideally go through the following points: (0, 20.5625) (10, 27.5625) (30, 14.5625) (50, 14.6875) (60, ...