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
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1answer
154 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
votes
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?
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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 * (...
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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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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 ...
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
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
votes
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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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
votes
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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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^...
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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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1answer
120 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
votes
1answer
105 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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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
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0answers
82 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 ...
0
votes
3answers
117 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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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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votes
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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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
votes
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?
11
votes
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
352 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 ...
9
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
votes
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 ...
8
votes
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
votes
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
votes
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 $...
7
votes
2answers
3k views

Division algorithm for multivariate polynomials?

We know that if $F$ is a field and $f(X)$ a non-zero polynomial in $F[X]$, then for every polynomial $g(X)$ we can find $q(X),r(X)$ such that $$g(X)=f(X)\cdot q(X)+r(X)$$ with $r(X)$ the zero ...
7
votes
4answers
423 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
3answers
737 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
votes
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
votes
2answers
328 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
116 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
5answers
249 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
3answers
3k views

Factoring third degree polynomials using long division

I am sure there is a better strategy that someone smarter than me would use, but I am not that person. I am trying to factor $$x^2 - 3x^2 - 4x + 12$$ I don't know how so I attempt to guess with ...
6
votes
1answer
94 views

What is $k_{\text{max}}$?

If $[1-\cos x][1 - \cos 2x][1 - \cos 3x] = k\ ; 0º < x < 90º$ Find $k_{\text{max}}$ I have no idea how to solve this I've got $8\left[\sin\left(\frac{x}{2}\right)\times\sin x \...
6
votes
4answers
69 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
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
4answers
317 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\...
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
3answers
2k views

What do three consecutive dots mean in algebra?

In the formula, $$\frac{n(n-1)(n-2)\cdots(n-r+2)}{(r-1)!}a^{n-r+1}b^{r-1}$$ what does the "$\cdots$" mean?
6
votes
2answers
460 views

An Interesting Question about Pythagorean Triples

I have recently thought about a interesting question about Pythagorean Triples. Consider such a right-angled trapezium formed by 3 right-angled triangle. Determine does it exist integral ...
5
votes
3answers
266 views

Help to understand proof to show that a function is uniformly continous in a certain interval (Spivak)

This excerpt relates to the following proof: If $f$ is continous on $[a,b]$, then $f$ is uniformly continuous $[a,b]$. For $\epsilon > 0$ let's say that $f$ is $\epsilon$-good on $[a,b]$ if ...
5
votes
3answers
206 views

Why is the binomial coefficient related to the binomial theorem?

The binomial coefficient basically provides the number of ways to choose a set of $k$ from $n$ sets. To me, it can be considered the number of unique ways to pick $k$ amount of "cards" from a deck of $...
5
votes
3answers
480 views

Physics: Uniform Motion

Ok so I have this homework problem. I dont want to give out the information because i want to put in the values myself, but basically I have one object moving at said speed. Said time later, another ...
5
votes
1answer
131 views

How can one find intermediate digits of a root of an algebraic equation?

I was wondering whether there is a way to find intermediate digits of an algebraic equation. For example, if I have $$234x^{\frac{1}{12345}}-24621x^{\frac{1}{3456}}=1$$ And I want to find the $10^9$...
5
votes
4answers
3k views

Finding double root. An easier way?

Given the polynomial $f = X^4 - 6X^3 + 13X^2 + aX + b$ you have to find the values of $a$ and $b$ such that $f$ has two double roots. I went about this by writing the polynomial as: $$f = X^4 - 6X^3 +...
5
votes
3answers
127 views

Is $|x-y|^n\leq 2^n(|x|^n+|y|^n)$?

Is $$|x-y|^n\leq 2^n(|x|^n+|y|^n)$$ for all $x,y\in\mathbb{R}$ and $n=1,2,3,\dots$ a standard inequality? If so, what's its name or how do you prove it?
5
votes
5answers
290 views

Calculate $x$, if $y = a \cdot \sin{[b(x-c)]}+d$

I am not an expert when it comes to trigonometric functions. I need to calculate value of $x$ for a program. If $y = a \cdot \sin{[b(x-c)]}+d$ then what will be the formula to calculate $x$? Please ...