# Tagged Questions

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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### 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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### 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?
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### 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.
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### 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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### 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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### 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. ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### Formula for sum of $n^n$

How can we find an equation for $S(n)$ where: $$S_n = \sum\limits_{i=1}^n i^{i} = 1^1 + 2^2 + \dots + n^n$$ Thanks in advance!
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 $\sigma_j^2$ $$\implies\frac{\sigma_j^2}{\sigma_\widehat e^... 2answers 63 views ### Some diffuculties trying to compute double sums I have the following sum$$\sum_{i = 0}^{n-2}\sum_{j=i}^{n}(i + j) + \sum_{i = 0}^{n-2}\sum_{j=i}^{n}1$$? and i have no idea how to continue from here? 3answers 56 views ### how to solve this: z^2-(1-3i)z-2i-2=0 I've tried two ways, but get stuck. I've tried to simplify, but didnt know what to do next, and i've tried to solve it like Quadratic equation, but got stuck too. tnx.. one way got me this: z/2 * (... 5answers 157 views ### Why is reminder of 8^{30} / 7 same as that of 1^{30} / 7 I am not able to figure out why the reminder of 8^{30} / 7 is same as that of 1^{30} / 7. I know Euclid division a=bq+r but I don't know modular arithmetic, so please explain without referring ... 1answer 176 views ### Johann Bernoulli did not fully understand logarithms? This wikipedia article makes the claim: "Bernoulli's correspondence with Euler (who also knew the above equation) shows that Bernoulli did not fully understand logarithms." This is found under "... 0answers 153 views ### forming ODE by elimination of arbitrary constant Let$$y= \sin (a)e^{2x}+e^{a+3x}+\ln(a)e^x$$If we differentiate , we get y'. Now, since the number of arbitrary constant is 1, we can expect the differential equation to be of order=1.But we are ... 2answers 56 views ### \sum_{k=1}^n(k!)(k^2+k+1) for n=1,2,3… and obtain an expression in terms of n Find a closed expression in terms of n.$$\sum_{k=1}^n(k!)(k^2+k+1); n=1,2,3...$$Any idea about how to do this.. I'm a new to this so a little explanation would be helpful. Thanks in advance! 2answers 2k views ### Proof for Binomial theorem I need to prove this general formula (1+x)^{n} = \sum_{k=0}^{n} \frac{n!}{k!(n-k)!}x^{k} And also prove to prove it on example - equivalence of (1+x)^{5} and its expansion 1+5\frac{5}{1!}x+..... 4answers 63 views ### Prove |a - b|< c if and only if b - c < a < b + c. Prove |a - b|< c if and only if b - c < a < b + c. It is a task from real analysis and I am failing the class I tried doing it on a quiz, but I got it incorrect. 5answers 1k views ### Can this function be rewritten to improve numerical stability? I'm writing a program that needs to evaluate the function$$f(x) = \frac{1 - e^{-ux}}{u}$$often with small values of u (i.e. u \ll x). In the limit u \to 0 we have f(x) = x using L'Hôpital's ... 5answers 360 views ### Subtracting expressions with radicals I want to subtract the expressions 20\sqrt{72a^3b^4c} - 14\sqrt{8a^3b^4c}. I simplified this to 120ab^2\sqrt{2ac}-28ab^2\sqrt{2ac}. My textbook says the answer is 92ab^2\sqrt{2ac}. Why doesnt ... 3answers 687 views ### How do I solve \vert x\vert^{x^2-2x} = 1? I have the exponential equation \vert x\vert^{x^2-2x} = 1, but how do I solve it? 4answers 23k views ### How to solve a quartic equation? Could someone please explain how to solve this : x^4 - 10x^3 + 21x^2 + 40x - 100 = 0 - not the answer only, but a step-by-step solution. I tried to solve it, with the help of khanacademy, but still ... 2answers 295 views ### Can someone explain this anecdote from Bob Weinstock? In this interesting essay explaining the performance gap among minorities in elite universities, there is an anecdote at the very bottom of the essay which intrigued me. Here's the screenshot: I ... 2answers 98 views ### Minimum of \sqrt{\dfrac{a}{b+c}}+\sqrt{\dfrac{b}{c+a}}+\sqrt{\dfrac{c}{a+b}} What is the minimum of$$f(a,b,c):=\sqrt{\dfrac{a}{b+c}}+\sqrt{\dfrac{b}{c+a}}+\sqrt{\dfrac{c}{a+b}}$$where a,b,c are positive real numbers? When a=b=c, we have f(a,b,c)=\dfrac{3}{\sqrt{2}}\... 2answers 1k views ### How can I describe the area between two ellipses? Given two ellipses that take up regions E_1 and E_2 in \mathbb{R^2}, with the following properties: Centers defined in the Cartesian coordinate system (c_1, 0) for E_1 and (c_2, 0) for ... 3answers 2k views ### How to check if a quadratic surd is a perfect cube? While trying to answer this question, I got stuck showing that$$\sqrt[3]{26+15\sqrt{3}}=2+\sqrt{3}$$The identity is easy to show if you already know the 2+\sqrt{3} part; just cube the thing. If ... 3answers 742 views ### How do I completely solve the equation z^4 - 2z^3 + 9z^2 - 14z + 14 = 0 where there is a root with the real part of 1. I would please like some help with solving the following equation:$$z^4 - 2z^3 + 9z^2 - 14z + 14 = 0$$All I know about the equation is that there is a root with the real part of 1. My approach ... 4answers 164 views ### How to find the solution for \frac{2x-3}{x+1} \leq 1? I have the following inequality:$$\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$$... 4answers 424 views ### Proof of Inequality using AM-GM I just started doing AM-GM inequalities for the first time about two hours ago. In those two hours, I have completed exactly two problems. I am stuck on this third one! Here is the problem: If a, b, ... 2answers 330 views ### Rationalizing the denominator 3 It is a very difficult question. How can we Rationalizing the denominator?$$\frac{2^{1/2}}{5+3*(4^{1/3})-7*(2^{1/3})}$$4answers 117 views ### Diophantine equation: (x-y)^2=x+y I have to solve the following equation: (x-y)^2=x+y, where x and y are non-negative integers. This equation has an infinite number of solutions, but how to prove that there exists a positive ... 8answers 123 views ### How to show that 6^n always ends with a 6 when n\geq 1 and n\in\mathbb{N} Is there a proof that for where n is a natural number$$6^n$$will end with a 6? I can understand conceptually that 6\cdot 6 ends with 6 and then multiplying that by 6 will still end with ... 6answers 373 views ### How to show that \frac{x^2}{x-1} simplifies to x + \frac{1}{x-1} +1 How does \frac{x^2}{(x-1)} simplify to x + \frac{1}{x-1} +1? The second expression would be much easier to work with, but I cant figure out how to get there. Thanks 5answers 254 views ### Solving the inequality (x^2+3)/x\le 4 This is the inequality$$\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} ...
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\...