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
3k views

Perpendicular line passing through the midpoint of another line

I have several $2d$ line segments. for example, if I take a one line segment having end points $(x_1, y_1)$ and $(x_2, y_2)$. Then, I want to make a perpendicular line which passes through the ...
2
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3answers
218 views

Can $n(n+1)2^{n-2} = \sum_{i=1}^{n} i^2 \binom{n}{i}$ be derived from the binomial theorem?

Can this identity be derived from the binomial theorem? $$n(n+1)2^{n-2} = \sum_{i=1}^{n} i^2 \binom{n}{i}$$ I tried starting from $2^n = \displaystyle\sum_{i=0}^{n} \binom{n}{i}$ and dividing it ...
2
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4answers
2k views

Factoring Cubic Equations

I’ve been trying to figure out how to factor cubic equations by studying a few worksheets online such as the one here and was wondering is there any generalized way of factoring these types of ...
2
votes
2answers
201 views

The Roots of Unity and the diagonals of the n-gon inscribed in the unit circle

I want to prove that the sum of the squares of the diagonals of a regular $n$-gon inscribed in the unit circle is equal to $2n$. So what I've done is I considered the $n$th roots of unity and said ...
2
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4answers
169 views

Fast square roots

I need to compute the square roots of lots of numbers. The numbers increase monotonically by fixed step. For example, 1, 2, 3, ..., 1 000 000. What is the fastest way to do so? Is it possible somehow ...
2
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8answers
241 views

Why does $\frac{a}{\frac{b}{x}} = x \times \frac{a}{b}$?

As much as it embarasses me to say it, but I always had a hard time understanding the following equality: $$ \frac{a}{\frac{b}{x}} = x \times \frac{a}{b} $$ I always thought that the left-hand side ...
2
votes
2answers
315 views

Rationalizing the denominator of $\frac {\sqrt{10}}{\sqrt{5} -2}$

I have the expression $$\frac {\sqrt{10}}{\sqrt{5} -2}$$ I can't figure out what to do from here, I can't seem to pull any numbers out of either of the square roots so it appears that it must remain ...
2
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3answers
219 views

Superball total bounce distance

I am asked to explain how to calculate total bounce distance: A "super" rubber ball is one that is measured to bounce when dropped 70% or higher that the distance from which it is dropped. You are ...
2
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3answers
1k views

Find the solution of this equation $(\sqrt{2-\sqrt 3})^x+(\sqrt{2+\sqrt 3})^x=4$ [duplicate]

Possible Duplicate: Solve $(\sqrt{5+2\sqrt{6}})^{x}+(\sqrt{5-2\sqrt{6}})^{x}=10$. Help me solve this equation $(\sqrt{2-\sqrt 3})^x+(\sqrt{2+\sqrt 3})^x=4$
2
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0answers
627 views

Reference books for highschool Algebra and Geometry?

I'm tutoring high school students in Math for a local College and Career prep program and would like to have a reference book on hand that I can consult. I'm a Comp Sci graduate so I have a pretty ...
2
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2answers
988 views

Supremum and infimum of $\{\frac{1}{n}-\frac{1}{m}:m, n \in \mathbb{N}\}$

I would like to verify my proof of the following: Let $A=\{\frac{1}{n}-\frac{1}{m}:m, n \in \mathbb{N}\}$. I want to show that $-1$ and $1$ are the infimum and supremum respectively. First I will ...
2
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2answers
162 views

Evaluating $\sum_{k=1}^n k^x$ [duplicate]

Possible Duplicate: Finite Sum of Power? Is there a general expression for $\sum_{k=1}^n k^x$ for any integer value of $x$? The table for $x=1,2,\dots 10$ is given here. Is there formula ...
2
votes
4answers
317 views

Calculating the chess problem, sum $\sum_{k=0}^{63}2^{k}$ [duplicate]

Possible Duplicate: Summation equation for $2^{x-1}$ I'm solving the classic problem of the inventor of chess, who according to legend sold the invention for one grain for the first square ...
2
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2answers
358 views

Does this qualify as a proof? (Spivak's 'Calculus')

I'm working through Spivak's 'Calculus' at the moment, and a question about series confused me a bit. I think I have the solution, but I'm not sure if my "proof" holds. The question is: Prove ...
2
votes
1answer
396 views

Composition of two polynomials

How's to make the composition of two polynomials? According to this page: If $ P = (x^3 + x) $, $ Q = (x^2 + 1) $ then, $ P\circ Q = P\circ (x^2 + 1) = (x^2 + 1)^3 + (x^2 + 1) = x^6 + 3 x^4 + 4 x^2 ...
2
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2answers
152 views

The roots of $t^5+1$

Just a quick question, how do we go about finding the roots of $t^5 +1$? I can see that since $t^5=-1$ that an obvious root is $\sqrt[5]{-1}$. I am assuming that since there is a $-1$ involved, some ...
2
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1answer
68 views

$n^s=(n)_s+f(s)$, what is $f(s)$?

In the following equation, $$n^s=(n)_s+f(s)$$ What is general form for $f(s)$? Understand that, $$(n)_s=n(n-1)(n-2)\cdots(n-[s-1])=\text{ The Falling Factorial }$$ I have experimented with this ...
2
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3answers
546 views

How to fill up the gap between a typical advanced undergraduate algebraic curve course and High school basic geometry/precalculus course?

Based on this question i asked recently: A question about geometry of plane curve books, i think it is too advance for a HS student/ typical second or third year undergraduate math majors to read on ...
2
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1answer
170 views

Find all reals $a, b$ for which $a^b$ is also real

The title is pretty much clear, but here is a more precise formulation: Find all pairs $(a,b)\in\mathbb{R^2}$ for which $a^b$ is also real. I used a CAS to solve the problem and it says that the ...
2
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4answers
7k views

Whats the rule for putting up a plus-minus sign when taking under root?

Given a simple equation.... $\ (x+1)^2 =21 $ if we take the under root of both sides , we get $\ x+1 = \pm \surd 21 $ why dont we get a $ \pm $ on the left hand side ?
2
votes
1answer
99 views

Expanding $(2y-2)^2$ by FOIL

Expanding $(2y-2)^2$ Isn't this same as $$(2y-2)(2y-2)\ ?$$ $$4y^2-6y+4$$ This should be FOILd shouldn't it?
2
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6answers
593 views

Next number in series

What are the basic/advanced strategies used to find the next number in series. I know the simple ones such as addition, multiplication etc. But recently I came into a question that goes on something ...
2
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3answers
263 views

How do I solve this equation involving a logarithm?

I'm running in circles and I don't understand how to do this. $$x\log(x) = 100$$ Where the $\log$ is in base $10$, I understand that $\log(y)=x$ is $10^x = y$. So is it the same for $x\log(x) = ...
2
votes
6answers
297 views

Factoring Quadratics

Is there a method to find which numbers to use when simplifying quadratics? For example $x^2 + 5x + 6$ is easy enough to find, but what if I have bigger numbers, or I have this quadratic expression: ...
2
votes
1answer
676 views

Interesting problem on “neighbor fractions”

This is from I. M. Gelfand's Algebra book. Fractions $\displaystyle\frac{a}{b}$ and $\displaystyle\frac{c}{d}$ are called neighbor fractions if their difference $\displaystyle\frac{ad - bc}{bd}$ ...
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4answers
673 views

Find remaining vertices of a square, given 2

I need a hint for this problem. Let the vertices of a square ABCD represent on the Argand diagram the complex numbers a,b,c, and d respectively. A,B,C,D are taken anti-clockwise in the order named. ...
2
votes
1answer
139 views

Solving inequalities comparing $f(x)$ to $0$ where $f$ is an elementary function

Any inequality comparing elementary functions can be rearranged to compare some elementary function $f$ to $0$. What is the best way to approach, in general, solving such inequalities at the ...
2
votes
1answer
301 views

How to rearrange formulas to calculate orbit from tangent and apoapsis

I need some help rearranging some orbital mechanics formulas. All images have been borrowed from http://www.braeunig.us/space/orbmech.htm which has a through treatment of orbital equations, but ...
2
votes
3answers
428 views

How is $Ax + By = C$ the equation of a straight line?

I know the equation, $y = mx + b$ where $m$ is slope and $b$ is $y$-intercept, is a straight line. But I know also that $Ax + By = C$ is a straight line equation, but how does it represent a straight ...
2
votes
2answers
314 views

Find the values of $m$ in the 2nd degree equation $mx^2-2(m-1)x-m-1=0$ so that it has one root between $-1$ and $2$

Find the values of $m$ in the 2nd degree equation $mx^2-2(m-1)x-m-1=0$ so that it has only one root between $-1$ and $2$. Like in this almost identical question there are two ways to solve this, one ...
2
votes
1answer
294 views

On Profit and loss

A dishonest dealer marks his goods $20\%$ above the cost price. He gives a discount of $10\%$ to the customer on the marked price and makes a profit by using a false weight of $900$ gms in place of ...
2
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4answers
169 views

How to sum up this series?

How to sum up this series : $$2C_o + \frac{2^2}{2}C_1 + \frac{2^3}{3}C_2 + \cdots + \frac{2^{n+1}}{n+1}C_n$$ Any hint that will lead me to the correct solution will be highly appreciated. EDIT: ...
2
votes
3answers
189 views

Are there publications or references describing N dimensional space divided by N-1 dimensional (hyper)planes?

As a teenager I was given this problem which took me a few years to solve. I'd like to know if this hae ever been published. When I presented my solution I was told that it was similar to one of ...
1
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1answer
53 views

Show the following inequality has exactly one integer solution

Problem We have the following inequality. Show that it has only one integer solution for each $n$. $$k^2+k-2\le2n\le k^2+3k-2$$ Attempt Solving this inequality, I got ...
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0answers
39 views

Polynomial With Complex Zeros

There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero of the polynomial $P(x) = x^3 - ax^2 + bx - 65$. For each possible combination of $a$ and $b$, let ...
1
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2answers
66 views

Given $r>0$, find $k>0$ such that $\sqrt{(x-2)^2+(y-1)^2}<k$ implies $|xy-2|<r $

Using the axioms, theorem, definitions of high school algebra concerning the real numbers, then prove the following: Given $r>0$, find a $k>0$ such that: $$\text{for all }x, y: ...
1
vote
1answer
67 views

Simplifying Sum

How would one show that $$ \sum_{i=0}^n\binom{n}{i}(-1)^i\frac{1}{m+i+1}=\frac{n!m!}{(n+m+1)!} ? $$ Any hint would be appreciated. Note: I tried to recognize some known formula, but since I don't ...
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2answers
61 views

how can I prove negative times negative is positive. [duplicate]

Well, I know the fact that negative times negative is positive. It would be interesting to question this fact and try to prove it.
1
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2answers
42 views

How do I solve an inequality with 2 inverse trigonometrical functions involved?

I haven't worked with this in a long time! All I remember is that increasing vs decreasing functions have the power of modifying the symbol. $$\arcsin\left(\dfrac{2}{x}\right) > ...
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0answers
134 views

Crossed Ladders Problem

Two ladders, one 10 meters long and the other 8 meters [long], have been placed in a trench as indicated in the opposite figure. Their point of intersection, M, is 3 meters from the base of the ...
1
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1answer
329 views

Exponential Growth and Decay / compound interest

This is the question: "If you want to have $\$75,000$ after $35$ years in your account that pays $12\%$ annual interest compounded quarterly, how much should you put in as your original investment?" ...
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2answers
86 views

simplify cos 1 degree + cos 3 degree +…+cos 43 degree?

I am currently working on a problem and reduced part of the equations down to $\cos(1^\circ)+\cos(3^\circ)+.....+\cos(39^\circ)+\cos(41^\circ)+\cos(43^\circ)$ How can I calculate this easily using ...
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1answer
102 views

Partial fraction decomposition on: $\frac{1}{(1-u^2)^2}$?

how does one perform Partial fraction decomposition on: $\large \frac{1}{(1-u^2)^2}$ ? the square at the denominator makes it a bit non-standard...
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2answers
50 views

Changing the cartesian coordinate system

The cartesian coordinate in 3D is given as: Are we allowed to make our own coordinate system (switching axes around). The question is can we change the axes around? ** Like: **LOOKING AT THE ...
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0answers
58 views

Find n that satisfies the following [duplicate]

Find the smallest positive integer n that satisfies the following: We can color each positive integer with one of n colors such that the equation w + 6x = 2y + 3z has no solutions in positive integers ...
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vote
1answer
55 views

How do I find the horizontal asymptote of $f(x)=\frac{\sin (x) }{x}$?

I can instantly see that there will be a vertical asymptote at $x=0$, however I am finding it quite a challenge to find a horizontal asymptote. I've drawn the graph and it seems as if the amplitude of ...
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4answers
553 views

Show that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq \frac{9}{a+b+c}$ for positive $a,b,c$ [duplicate]

Show that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq \frac{9}{a+b+c}$, if $a,b,c$ are positive. Well, I got that $bc(a+b+c)+ac(a+b+c)+ab(a+b+c)\geq9abc$.
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1answer
43 views

Simplify this fraction with square roots; application to arctangent equation.

I need your help. I don't know how to simplify: $\frac{-1+\sqrt{3}+\sqrt{4+2\sqrt{3}}}{2\sqrt{3}} $ and $\frac{-1+\sqrt{3}-\sqrt{4+2\sqrt{3}}}{2\sqrt{3}}$ Thank you in advance. I found $1$ and ...
1
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3answers
65 views

How do I factor $\ t^4-2 \ $?

This binomial is part of a bigger problem that I need to solve, however, I am little stuck on how to factor it. $(t-1)(t-1)(t+1)$ does not work.
1
vote
2answers
39 views

Prove the given condition from given two quadratic equation

Question: If the quadratic equations $x^2+bx+c=0$ and $bx^2+cx+1=0$ have a common root then prove that either $b + c + 1 = 0$ or $b^2 + c^2 + 1 =bc + b + c$ Till yet, I had figured the common ...