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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0answers
37 views

Prove the identity $\tanh(N\textrm{acosh}\;a) = \vert \frac{g^{2N}-1}{g^{2N}+1}\vert$

During my recent study, I found an Identity which is of the form $$ \tanh(N\textrm{acosh}\;a) = \left\vert \frac{g^{2N}-1}{g^{2N}+1}\right\vert $$ where $a\geq1$ and $g>0$ satisfy $a=\frac{g^2+1}{...
1
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2answers
59 views

How can I prove that the follow polynomial is irreducible in $\mathbb{Q}$?

How can I prove that $x^5 + 7x^4 + 2x^3 + 6x^2 - x + 8$ is irrudicible in $\mathbb{Q}$? I can't use the Eisenstein's criterion and I tryed to put this polynomial in $\mathbb{Z}_3$ and $\mathbb{Z}_5$. ...
1
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4answers
682 views

Parabola and Circle problem : The parabola $y =x^2-8x+15$ cuts the x axis at P and Q. A circle is drawn …

Problem : The parabola $y=x^2-8x+15$ cuts the x axis at P and Q. A circle is drawn through P and Q so that the origin is outside it. Find the length at a tangent to the circle from O. My approach :...
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2answers
97 views

How to find the sum of the series by treating deonominator so that to split fraction $\frac{1}{a_1a_2a_3} + \frac{1}{a_2a_3a_4}+$…

This is a series in A.P ( Arithmetic Progression ) $\frac{1}{a_1a_2} + \frac{1}{a_2a_3}+\frac{1}{a_3a_4}+.......\frac{1} { a_{n}a_{n+1}}$ ( where $a_1 ,a_2,a_3.....$ are terms in A.P.) When we do ...
0
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2answers
5k views

Simple way for solving generic work and time problems

I was looking for a general way of formulating solutions for work and time problems. For example, 30 soldiers can dig 10 trenches of size 8*3*3 ft in half a day working 8 hours per day. How many ...
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3answers
120 views

Find the product of positive roots of equation $\sqrt{2008}\,x^{\log_{2008}x}=x^2$

Problem : Find the product of positive roots of equation $\sqrt{2008}\, x^{\log_{2008}x}=x^2$ Solution : The given equation can be written as $\sqrt{2008} \, x^{\log_{2008}x}=x^2 $ $\implies\...
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2answers
81 views

Can anyone help me find an $x$ for which $\sin x=-1/2$ and $\sin x=\sqrt{2}/2$?

I know that $\sin x=0$ when $x$ is of the form $x=n\pi$ for $n\in\mathbb{Z}$. But, I can't figure out an $x$ for which $\sin x=-1/2$ and $\sin x=\sqrt{2}/2$ are both true. Can anyone help me?
0
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2answers
56 views

ordered triplets of integer $(x,y,z)$ in $z!=x!+y!$

$(1)\;:$ How many ordered triplets of positive integers $(x,y,z)$ are there are such that $z! = x!+y!$ $(2)\;:$ How many ordered triplets of positive integers $(x,y,z)$ are there are such that $w! =...
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2answers
2k views

Defining the width of a Gaussian function

I have the following Gaussian function: $$\rho(r) = q_i (\alpha/\pi)^{3/2} \exp(-\alpha r^2)$$ Qualitatively, the "width" of this Gaussian is related to $\frac{1}{\alpha}$: the larger the value of $\...
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2answers
2k views

How do calculators handle $\pi$?

When the calculator displays the digits of $\pi$, how does it arrive at that answer? Also, at what digit does the approximation of $\pi$ stop at?
0
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4answers
162 views

How can I factor the polynomial $125x^3 + 216$?

$$125x^3 + 216$$ I have tried to factor it but because the square root of $216$ is a decimal, I can't figure out how to do the problem.
0
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2answers
107 views

Calculate Points for a Parallel Line

Given a line running through p1:(x1,y1) and p2:(x2,y2), I need to calculate two points such that a new parallel line 20 pixels away from the given line runs through the two new points. Edit: The ...
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3answers
123 views

A non-zero polynomial with real coefficients has the property that $f(x)=f'(x).f''(x)$.Then find the leading coefficient of $f(x).$

A non-zero polynomial with real coefficients has the property that $f(x)=f'(x).f''(x)$.Then find the leading coefficient of $f(x).$ I let $f(x)=a_0x^n+a_1x^{n-1}+a_2x^{n-2}+.....+a_n$ and then i ...
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1answer
109 views

What justifies algebraic manipulation in equations with only variables?

I recognise my question is at a beginner level but my current level of knowledge of math is up to what any undergraduate engineer would know, so you can give me a more-than-beginner-level explanation. ...
0
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2answers
110 views

How to solve $\frac{2}{3\sqrt{2}}=\cos\left(\frac{x}{2}\right)$?

How do you solve $\dfrac{2}{3\sqrt{2}}=\cos\left(\dfrac{x}{2}\right)$ for $x$ in the interval $0 \leq x \leq 2\pi$? This comes from a question that I asked before. I frequently get stumped when ...
0
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1answer
71 views

Powers of $x$ which are always positive

I've learnt that the square of any real number is always positive. So, we know $x^2\ge0$ for any real number $x$. Similarly, $x^{2k}\ge0$ for any $x\in\mathbb R$ and $k\in\mathbb N$. How can we find ...
0
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1answer
101 views

Exponential equation+derivative

I saw here on math.stackexchange.com an equation which has very nice solutions (by solutions I mean a proof): $3^x+28^x=8^x+27^x$, where $x$ is a real number. However, I think there must be an ...
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7answers
941 views

Perpendicular bisector

Show that BE is the perpendicular bisector to AC. I tried to prove this through Pythagoras, but the answer I got did not prove it was at a right angle, and therefore said it was not the ...
0
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1answer
123 views

Formula for sum of $n^n$

How can we find an equation for $S(n)$ where: \begin{equation} S_n = \sum\limits_{i=1}^n i^{i} = 1^1 + 2^2 + \dots + n^n \end{equation} Thanks in advance!
0
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2answers
50 views

How is $ [(x+h)^{1/3} - x^{1/3}] [(x+h)^{2/3} +x^{1/3}(x+h)^{1/3}+ x^{2/3}] $ simplified to become $ (x+h-x) $?

How is $ [(x+h)^{1/3} - x^{1/3}] [(x+h)^{2/3} +x^{1/3}(x+h)^{1/3}+ x^{2/3}] $ simplified to become $ (x+h-x) $ ?? I'm currently reading a text and I've been trying to get the hang of this for a ...
0
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2answers
2k views

Root of a polynomial with rational coefficients

I am currently learning about Direct Proofs. I am struggling trying to find a starting point to prove the Statement: For all real numbers $c$, if $c$ is a root of a polynomial with rational ...
0
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1answer
155 views

A curve such that all lines on the plane intersect it : cont..

Further to this question (which appears more or less settled); "Is there a curve on plane such that any line on the plane meets it (a non zero ) finite times ?" I ask now the upper bounds of the ...
0
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3answers
1k views

Find the maximum value of the product xyz?

IF $x , y , z$ are arbitary positive real numbers satisfying the equation $$ 4xy + 6yz + 8xz = 9$$ Find the maximum value of the product $xyz$. I dont know from where to begin . 3 variables and ...
0
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2answers
44 views

Simple summation problem regarding origin of summand:

If $$\frac{1}{\sigma_\widehat{e}^2}=\sum_i\frac{1}{\sigma_i^2}\tag{1}$$ Pick any one of the $\sigma_j$ and multiply both sides of $(1)$ by $\sigma_j^2$ $$\implies\frac{\sigma_j^2}{\sigma_\widehat e^...
0
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2answers
63 views

Some diffuculties trying to compute double sums

I have the following sum $$\sum_{i = 0}^{n-2}\sum_{j=i}^{n}(i + j) + \sum_{i = 0}^{n-2}\sum_{j=i}^{n}1$$? and i have no idea how to continue from here?
0
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3answers
56 views

how to solve this: $z^2-(1-3i)z-2i-2=0$

I've tried two ways, but get stuck. I've tried to simplify, but didnt know what to do next, and i've tried to solve it like Quadratic equation, but got stuck too. tnx.. one way got me this: z/2 * (...
0
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5answers
157 views

Why is reminder of $8^{30} / 7$ same as that of $1^{30} / 7$

I am not able to figure out why the reminder of $8^{30} / 7$ is same as that of $1^{30} / 7$. I know Euclid division $a=bq+r$ but I don't know modular arithmetic, so please explain without referring ...
0
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1answer
176 views

Johann Bernoulli did not fully understand logarithms?

This wikipedia article makes the claim: "Bernoulli's correspondence with Euler (who also knew the above equation) shows that Bernoulli did not fully understand logarithms." This is found under "...
0
votes
0answers
153 views

forming ODE by elimination of arbitrary constant

Let $$y= \sin (a)e^{2x}+e^{a+3x}+\ln(a)e^x$$ If we differentiate , we get $y'$. Now, since the number of arbitrary constant is 1, we can expect the differential equation to be of order=1.But we are ...
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2answers
56 views

$\sum_{k=1}^n(k!)(k^2+k+1)$ for $n=1,2,3…$ and obtain an expression in terms of $n$

Find a closed expression in terms of $n$. $$\sum_{k=1}^n(k!)(k^2+k+1); n=1,2,3...$$ Any idea about how to do this.. I'm a new to this so a little explanation would be helpful. Thanks in advance!
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2answers
2k views

Proof for Binomial theorem

I need to prove this general formula $(1+x)^{n} = \sum_{k=0}^{n} \frac{n!}{k!(n-k)!}x^{k}$ And also prove to prove it on example - equivalence of $(1+x)^{5}$ and its expansion $1+5\frac{5}{1!}x+.....
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4answers
63 views

Prove $|a - b|< c$ if and only if $b - c < a < b + c$.

Prove $|a - b|< c$ if and only if $b - c < a < b + c$. It is a task from real analysis and I am failing the class I tried doing it on a quiz, but I got it incorrect.
11
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5answers
1k views

Can this function be rewritten to improve numerical stability?

I'm writing a program that needs to evaluate the function $$f(x) = \frac{1 - e^{-ux}}{u}$$ often with small values of $u$ (i.e. $u \ll x$). In the limit $u \to 0$ we have $f(x) = x$ using L'Hôpital's ...
11
votes
5answers
360 views

Subtracting expressions with radicals

I want to subtract the expressions $20\sqrt{72a^3b^4c} - 14\sqrt{8a^3b^4c}$. I simplified this to $120ab^2\sqrt{2ac}-28ab^2\sqrt{2ac}$. My textbook says the answer is $92ab^2\sqrt{2ac}$. Why doesnt ...
11
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3answers
687 views

How do I solve $\vert x\vert^{x^2-2x} = 1$?

I have the exponential equation $\vert x\vert^{x^2-2x} = 1$, but how do I solve it?
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votes
4answers
23k views

How to solve a quartic equation?

Could someone please explain how to solve this : $x^4 - 10x^3 + 21x^2 + 40x - 100 = 0$ - not the answer only, but a step-by-step solution. I tried to solve it, with the help of khanacademy, but still ...
8
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2answers
295 views

Can someone explain this anecdote from Bob Weinstock?

In this interesting essay explaining the performance gap among minorities in elite universities, there is an anecdote at the very bottom of the essay which intrigued me. Here's the screenshot: I ...
8
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2answers
98 views

Minimum of $\sqrt{\dfrac{a}{b+c}}+\sqrt{\dfrac{b}{c+a}}+\sqrt{\dfrac{c}{a+b}}$

What is the minimum of $$f(a,b,c):=\sqrt{\dfrac{a}{b+c}}+\sqrt{\dfrac{b}{c+a}}+\sqrt{\dfrac{c}{a+b}}$$ where $a,b,c$ are positive real numbers? When $a=b=c$, we have $f(a,b,c)=\dfrac{3}{\sqrt{2}}\...
8
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2answers
1k views

How can I describe the area between two ellipses?

Given two ellipses that take up regions $E_1$ and $E_2$ in $\mathbb{R^2}$, with the following properties: Centers defined in the Cartesian coordinate system $(c_1, 0)$ for $E_1$ and $(c_2, 0)$ for $...
8
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3answers
2k views

How to check if a quadratic surd is a perfect cube?

While trying to answer this question, I got stuck showing that $$\sqrt[3]{26+15\sqrt{3}}=2+\sqrt{3}$$ The identity is easy to show if you already know the $2+\sqrt{3}$ part; just cube the thing. If ...
7
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3answers
742 views

How do I completely solve the equation $z^4 - 2z^3 + 9z^2 - 14z + 14 = 0$ where there is a root with the real part of $1$.

I would please like some help with solving the following equation: $$z^4 - 2z^3 + 9z^2 - 14z + 14 = 0$$ All I know about the equation is that there is a root with the real part of $1$. My approach ...
7
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4answers
164 views

How to find the solution for $\frac{2x-3}{x+1} \leq 1$?

I have the following inequality: $$\frac{2x-3}{x+1}\leq1$$ so, considering $x \neq -1$, I started multiplying $x+1$ both sides: $$2x-3\leq x+1$$ then I subtracted $x$ both sides: $$x-3\leq1$$ ...
7
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4answers
424 views

Proof of Inequality using AM-GM

I just started doing AM-GM inequalities for the first time about two hours ago. In those two hours, I have completed exactly two problems. I am stuck on this third one! Here is the problem: If $a, b, ...
7
votes
2answers
330 views

Rationalizing the denominator 3

It is a very difficult question. How can we Rationalizing the denominator? $$\frac{2^{1/2}}{5+3*(4^{1/3})-7*(2^{1/3})}$$
7
votes
4answers
117 views

Diophantine equation: $(x-y)^2=x+y$

I have to solve the following equation: $(x-y)^2=x+y$, where $x$ and $y$ are non-negative integers. This equation has an infinite number of solutions, but how to prove that there exists a positive ...
6
votes
8answers
123 views

How to show that $6^n$ always ends with a $6$ when $n\geq 1$ and $n\in\mathbb{N}$

Is there a proof that for where $n$ is a natural number $$6^n$$ will end with a $6$? I can understand conceptually that $6\cdot 6$ ends with $6$ and then multiplying that by $6$ will still end with $...
6
votes
6answers
373 views

How to show that $\frac{x^2}{x-1}$ simplifies to $x + \frac{1}{x-1} +1$

How does $\frac{x^2}{(x-1)}$ simplify to $x + \frac{1}{x-1} +1$? The second expression would be much easier to work with, but I cant figure out how to get there. Thanks
6
votes
5answers
254 views

Solving the inequality $(x^2+3)/x\le 4$

This is the inequality $$\left(\frac {x^2 + 3}{x}\right) \le 4 $$ This is how I solve it The $x$ in the left side is canceled and $4x$ is subtracted from both sides. $$\not{x} \left (\frac {x^2+3} ...
6
votes
4answers
74 views

Why do extraneous solutions exist?

I am currently in a Pre Calculus class at my High School. I have come across the concept of extraneous solutions, particularly when solving absolute value equations, radical equations, and logarithmic ...
6
votes
4answers
318 views

Trig equation help please

I am trying to solve $\sqrt{3}\tan\theta=2\sin\theta$ on the interval $[-\pi,\pi]$. $$\sqrt{3}\tan\theta=2\sin\theta \Rightarrow \sqrt{3}=\frac{2\sin\theta}{\tan\theta}$$ $$\Rightarrow \sqrt{3}=2\...