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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4
votes
2answers
107 views

Getting a standard form quadratic from a set of points ($3$ points)

I came home from school today, pulled out my homework, now I'm stumped. I don't want the answer, I just want to know how to do it. Here is the question that I'm reading: Determine a quadratic ...
0
votes
2answers
103 views

Simplifying Quadratic Equation

Quick question, say I'm simplying a solution I got using the quadratic equation and I run into this: Original version (as posted by OP): x = -7 +- 3 sqrt(5) over 3 Edited version: $$ x = ...
4
votes
2answers
277 views

Which is the “fastest” paper-pencil method to compare $\sqrt[17]{6}$ and $\sqrt[16]{4}$?

Which is the "fastest" paper-pencil method to compare (find which one is greater) $\sqrt[17]{6}$ and $\sqrt[16]{4}$? My analysis bought this whole thing down to comparing which is greater ...
0
votes
2answers
193 views

Row-sum for an integer triangle w/o brute force

The positive integers are arranged in increasing order in a triangle, as shown*. Each row contains one more number than the previous row. The sum of the numbers in the row that contains the number 400 ...
1
vote
3answers
138 views

Trying to solve this simple algebra problems: $\frac{5 + 8x}{3 + 2x} = \frac{45 - 8x}{13 - 2x}$

I know it's kind of stupid to ask this question. But I have problems to solve this simple problems. Can someone point me to the right direction? Did I do something wrong in the process or it's a ...
2
votes
1answer
857 views

Finding points which divides a right trapezoid's area into equal pieces

I have a right trapezoid as follows; We have $h$, $b$ and $a$. For any $n$, I need to divide total area of trapezoid into equal parts. I have to find a general formulation for the length of $p$ ...
10
votes
2answers
677 views

Proof of an inequality: $\sqrt{n} < \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}}$ [duplicate]

Possible Duplicate: Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction How do I prove the following? $$\sqrt{n} < \dfrac{1}{\sqrt{1}} + \dfrac{1}{\sqrt{2}} + ...
2
votes
1answer
277 views

Need help with the proof of conic section

Prove that the intersection of a plane and a object consist of one cone and one upside-down cone where the tip of cone meet is either degenerate conic or conic Also, idenify in what situation, the ...
0
votes
1answer
195 views

Solution for following trigonometric equation?

I have the following trigonometric equation in $\theta$: $$0=G_{\omega}(1/r^2)({\csc^2}\theta){(r\cos\theta-x)}^2+(\cot\theta)(r\cos\theta-x)+r\sin\theta-y.$$ Is there an analytical solution for ...
0
votes
1answer
143 views

finding the decay constant

Given the following function, how does one rewrite the exponential part of the equation into $e^{-L/L_{0}}$, where $L_{0}$ is the decay constant ...
1
vote
1answer
133 views

How to control the range in a reciprocal function

Given the reciprocal function $$\frac{a}{m \cdot x + b}$$ where $a,m,b$ are constants. I'm trying to figure out how/if I can control the range that this produces. The application of this problem is ...
0
votes
1answer
69 views

What is $x$ in the following question?

Here is the question: $$ \frac 23 \div x = \frac 12 + \frac 13 .$$ I would be very grateful if someone could find the value of $x$. Thank You.
2
votes
2answers
127 views

Geometric series

Suppose $$X = 1.05^{35}v+1.05^{34}v^{2} + \cdots + 1.05v^{35}$$ where $v = 1/1.05$. Then we have $$X = 1.05^{35}v(1+ 0.952v+ \cdots + 0.952^{34}v^{34})$$ So the sum of this would be $$ X = ...
0
votes
2answers
640 views

Calculating the meeting point of these two runners

I am reading about an algorithm. For a part of it, to present the underlying logic uses the following analogy which I do not understand: If two people run on a track and one of them runs with ...
3
votes
2answers
218 views

Completing the square

How might I find linear combinations $$\begin{align*} A&=a_1x+a_2y+a_3z\\ B&=b_1x+b_2y+b_3z\\ C&=c_1x+c_2y+c_3z \end{align*}$$ Such that I can transform the two polynomials ...
2
votes
3answers
343 views

How to find $n$'th term of the sequence $3, 7, 12, 18, 25, \ldots$?

$$3, 7, 12, 18, 25, \ldots$$ This sequence appears in my son's math homework. The question is to find the $n$'th term. What is the formula and how do you derive it?
3
votes
2answers
152 views

Prove by induction that $2^1+2^2+2^3+2^4+ \cdots +2^n=2(2^n-1)$

Alright I have this problem, Prove by induction $2^1+2^2+2^3+2^4+ \cdots +2^n=2(2^n-1)$ Now I've done this so far: Base case $n=1$: $$2^1 = 2$$ $$2(2^1-1)=2(2-1)=2(1)=2 .$$ Assume for $k$, prove ...
0
votes
1answer
277 views

How to explain this $3=2$ proof? [duplicate]

Possible Duplicate: The $3 = 2$ trick on Google+ I saw google some links http://www.astahost.com/info/tiiiss-ramanujams-proof-flaws.html $$-6 = -6 $$ $$9-15 = 4-10 $$ adding ...
51
votes
4answers
3k views

Are the solutions of $x^{x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}}}=2$ correct?

Problem: Find $x$ in $$\large x^{x^{x^{x^{ \cdot^{{\cdot}^{\cdot}} }}}}=2$$ Trick: $x^{x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}}}=2$, so, $x^{(x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}})}=x^2=2$, and, ...
3
votes
2answers
648 views

What's a good progression/path for self-educating from pre-algebra and beyond?

I never got very far in my math studies; I took zero math in college and had terrible teachers in high school. Fast forward to today and I'm a professional developer who's starting to feel that my ...
0
votes
1answer
138 views

Inequality question (with complex numbers)

I was wondering how would you derive/get $|q(z)|\ge R^2-|a|R-|b|>R^2/2 $? Thanks.
0
votes
1answer
81 views

Changing order of sums

I would like to change the order so that I know the coefficients of each $x^i$. How do I change the order of the summation in the following sum? $$ ...
2
votes
3answers
711 views

Does anyone know l'Hôpital's rule for limits?

I was doing an assignment and the middle 2 questions are l'Hôpital's rule questions and we haven't even done this in class yet. These are 1 or 2 chapters away yet we are expected to do them and I have ...
3
votes
2answers
194 views

Is this intermediate value theorem or extreme value theorem?

I cant understand how to prove this question. We learned about intermediate value theorem but this makes no sense because $120$ km isn't in bounds of either upper or lower limit. Here is the question ...
2
votes
1answer
311 views

How can I solve a system of equations?

If $x, y, z$ are complex numbers, how can I solve this system of equations \begin{cases} x(x-y)(x-z)=3;\ \\y(y-z)(y-x)=3;\ \\z(z-x)(z-y)=3. \end{cases}
7
votes
3answers
449 views

Hard elementary combinatorics problem

How does one compute (without brute force) the smallest integer $n$ such that $\binom{2n}{1}(-3)^0 + \binom{2n}{3}(-3)^1 + \binom{2n}{5}(-3)^2 + \cdots + \binom{2n}{2n-1}(-3)^{(n-1)} = 0$?
3
votes
2answers
698 views

using squeeze theorem to show limit

I have a function $g(x)=x^4 \cos(2/x)$. I have to use Squeeze Theorem to show $\lim\limits_{x\to0}g(x)=0$. Usually all the questions I have done up till now on Squeeze theorem have provided me with ...
1
vote
2answers
201 views

How to find the minimum value of $px+qy$ when $xy=r^2$?

The question says: "Find the minimum value of $px+qy$ when $xy=r^2$." No information is given on $p,q,x,\text{and }y.$ However assuming the obvious I tried using this, but I am not able reduce it to ...
5
votes
4answers
2k views

Example of a real-life graph with a “hole”?

Anyone ever come across a real non-textbook example of a graph with a hole in it? In Precalc, you get into graphing rational expressions, some of which reduce to a non-rational. The cancelled ...
1
vote
2answers
701 views

bird traveling to a nest wants to save energy

This is a multiple choice question in one of tests I just wrote and I did not know the answer to it. I was just stuck on this during the test. It is a very weird question, one I find to be impossible. ...
1
vote
2answers
208 views

Upper and lower bounds in regards to 0.(9) [duplicate]

Possible Duplicate: Does .99999… = 1? I'm only doing this at GSCE and I'm really only asking here because of an interesting email conversation between my Grandfather and I regarding ...
3
votes
3answers
92 views

Determine if the equation is valid/true

The equation is: $$\log_b \frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}} = 2\log_b(\sqrt{3}+\sqrt{2}).$$ I can get as far as: $$\log_b(\sqrt{3}+\sqrt{2}) - \log_b(\sqrt{3}-\sqrt{2}) = ...
1
vote
4answers
186 views

the sum of powers of $2$ between $2^0$ and $2^n$

Lately, I was wondering if there exists a closed expression for $2^0+2^1+\cdots+2^n$ for any $n$?
0
votes
2answers
713 views

Express the logarithm in terms a,b,c

Suppose that: $\log_{10}A = a$ $\log_{10}B = b$ $\log_{10}C = c$ I need to express the following in terms of $a$,$b$,$c$. $\log_{10}A + 2\log_{10}(1/A)$ $\log_{10}(((AB)^5)/C)$ ...
2
votes
2answers
77 views

Evaluate expression using change of base

This is an awkward question to me, it was not covered in class. Suppose: $$\begin{align*} \log_b 2 &= A, \\ \log_b 3 &= B, \\ \log_b 5 &= C. \end{align*} $$ Then, use the ...
3
votes
2answers
3k views

Expressing as a single logarithm

I've got the equation: $$\log_{10}(x^2 - 16) - 3\log_{10}(x + 4) + 2\log_{10} x$$ I'm looking to express this as a single logarithm. I came up with $$\log_{10}(x^2 - 16) - \log_{10}(x + 4)^3 + ...
3
votes
2answers
356 views

How many times more than $0$?

If I have $10$ apples, but you have $5$ apples, then I have $2$ times more apples than you. But what if I have $10$ apples, but you don't have any apples? If you look at the graph ...
-1
votes
1answer
431 views

change of order in a double series

Sorry for this simple question, but I never learned nothing about double sums )= , I only know what they are . But finally I had a problem where I need to know something about this, wondering if ...
5
votes
6answers
9k views

How do you factor $x^3-3x^2+3x-1$?

$$x^3-3x^2+3x-1?$$ I know this may seem trivial, but I, for the life of me, I cannot figure out how to factor this polynomial, I know that the root is $$(x-1)^3=0$$ because of wolframalpha, but I ...
2
votes
5answers
816 views

How do you divide a polynomial by a binomial of the form $ax^2+b$, where $a$ and $b$ are greater than one?

I came across a question that asked me to divide $-2x^3+4x^2-3x+5$ by $4x^2+5$. Can anyone help me?
0
votes
1answer
129 views

Help on understanding the proof for the max subsequence problem

I am trying to understand a proof. The problem is the detection of the maximum continuous subsequence sum problem (i.e. find the subsequence of an array that gives the maximum sum). The ...
1
vote
2answers
80 views

finding the value of an equation?

$f(x) = \log x$ for any real number $x > 0$ and $$g(n)=\begin{cases} n& \text{if $n$ is even}\\ \tfrac{1}{n}& \text{if $n$ is odd}.\end{cases}$$ for any natural number $n$. If $x$ is a ...
0
votes
3answers
5k views

How to prove whether a polynomial function is even or odd

We know that a function is even if $f(-x) = f(x)$ and odd if $f(-x) = -f(x)$. With this reasoning is it possible to prove that a polynomial function such as $f(x) = a_{2n}x^{2n} + a_{2n-2}x^{2n-2} + ...
3
votes
2answers
455 views

Solve the equations $\tan x=x$ and $\ln x=x$

Is it possible to solve the equations $\tan x=x$ and $\ln x=x$ in their respective domains of definition ?
8
votes
3answers
453 views

Prove that $\sum\limits_{k=1}^n \frac{1}{k^2+3k+1}$ is bounded above by $\frac{13}{20}$

I want ask a question about a sum. The exercise is as follows: Prove the following inequality for every $n \geq 1$: $$\sum\limits_{k=1}^n \frac{1}{k^2+3k+1} \leq \frac{13}{20} .$$
1
vote
2answers
276 views

Use division instead of multiplication

Is there any way to calculate x/y without using division? Like 1/10 = 1 * 0.1 . I'm asking because dividing is slower than multiplying in programming programs.
0
votes
2answers
258 views

Counting functions

How many functions are possible from the set $A=\{0,1,2\}$ into the set $B =\{0,1,2,3,4,5,6,7\}$ such that $f(i) \le f(j)$ for $i \lt j$ and $i,j \in A$? I am not sure which counting model ...
0
votes
2answers
99 views

simplify $a=a*b$

Normally in algebra, you can simplify an equation such as $a=a*b$ by dividing both sides by a common coefficient. For example, $\frac {a}{a}=\frac {ab}{a} \rightarrow1=b$. Obviously $b = 1$ is a ...
45
votes
4answers
2k views

A new imaginary number? $x^c = -x$

Being young, I don't have much experience with imaginary numbers outside of the basic usages of $i$. As I was sitting in my high school math class doing logs, I had an idea of something that would ...
1
vote
1answer
281 views

How to solve for the equation $ax \exp(bx)=c$?

How to solve for the equation $ax \exp(bx)=c$? It is known that $x\geq 0$.