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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6
votes
3answers
136 views

Sum of the $11^\mathrm{th}$ power of the roots of the equation $x^5+5x+1=0$

Find the sum of the $11^\mathrm{th}$ power of the all roots of the equation $$ x^5+5x+1=0 $$ My Attempt: Let $R=\{\alpha,\beta,\gamma,\delta,\mu\}$ be the set of all roots of the equation ...
0
votes
0answers
39 views

find gcd$( \lfloor x \rfloor, \lfloor x^a \rfloor)$

Is there a way to find gcd$( \lfloor x \rfloor, \lfloor x^a \rfloor)$ assume that $x>1$ and can assume that $a>0$. Also, if close form doesn't exist are there meaningful lower bounds and upper ...
1
vote
1answer
42 views

Circumference of separate circle

So I have been out of Algebra for a while now. I am trying to help my wife prep for an entrance exam and we ran across this in the practice test: ...
0
votes
2answers
49 views

integrating $\ln(ax)$ in an equation.

The derivative $\frac{d}{dx}\ln{(ax)} = \frac{1}{x}$ What follows is that $\int{\frac{d}{dx}\ln{(ax)}} = \int{\frac{1}{x}}$ And so, $\ln{(ax)} + c_1 = \ln{|x|} + c_2$ where $a, c_1, c_2 ...
3
votes
2answers
37 views

what geometric object is represented (in the complex plane) by the solution of an equation?

The solution to the equation: _ z = 2/z can be described as a geometric object, which? anyone know how to go about this problem? thanks in advance ...
2
votes
1answer
110 views

Minimizing the expression $(1+1/x)(1+m/y)$ over positive reals such that $mx+y=1$

Let $x$ and $y$ be positive real numbers such that $mx+y=1$. Find the positive $m$ such that the minimum of: $$\left( 1 + \frac{1}{x} \right)\left( 1 + \frac{m}{y} \right).$$ is $81$. I have ...
0
votes
1answer
56 views

How can I rearrange $Y =\frac{X}{A+BX}$ to solve for $X$?

I know Y, A, and B but how can I solve this for X? $Y =\dfrac{X}{A+BX}$ The $X$ value is the same number if that matters. I used this equation to solve for $A$ but I want to know how I can plug back ...
0
votes
2answers
139 views

Equating functions: does f=g?

$f(x)=\frac{x^2-2x}{x-2}$ $g(x)=x$ Does $f=g$? I said yes but my homework said they aren't equal.
1
vote
3answers
143 views

Is there any way to express $\theta=c$ as some function of $r$?

I recently found this: Desmos Graphing calculator. I tried to plot the equation $\theta=45$ but it gave me an error: Sorry, you can't graph $\theta$ as a function of anything yet. So I started ...
0
votes
1answer
50 views

Not understanding the solution to this rational expression

$$\sqrt { 1+\left(\frac { x }{ \sqrt { 1-{ x }^{ 2 } } } \right)^{ 2 } } $$ I have done the following: $$\sqrt { 1+ \frac { x^2 }{ { 1-{ x }^{ 2 } } } } $$ $$\sqrt{ \frac { 1-x^{ 2 } }{ 1-x^{ 2 ...
4
votes
1answer
58 views

Stuck simplifying a fractional expression

$$ \frac { \frac { 1 }{ 1+x+h } -\frac { 1 }{ 1+x } }{ h } $$ $$ \frac { 1(1+x) }{ 1+x+h(1+x) } -\frac { 1(1+x+h) }{ 1+x(1+x+h) } $$ $$ \frac { -h }{ (1+x+h)(1+x) } \quad *\quad \frac { 1 }{ h } $$ ...
0
votes
2answers
336 views

How would I simplify this compound fractional expression?

!http://imgur.com/stORiYu $$\frac{x^{-2}-y^{-2}}{x^{-1}+y^{-1}}$$ I know that a fractional expression with negative exponents can be flipped to get the positive exponent but I am not sure that I am ...
-1
votes
1answer
31 views

Reduction in profit…by how much

If an organization sells $n$ tickets then the profit would be 20% more than the total costs of production. Lets say that it sold all the tickets except 5% of them. What is the reduction in profit? ...
8
votes
3answers
3k views

Direct formula for area of a triangle formed by three lines, given their equations in the cartesian plane.

I read this formula in some book but it didn't provide a proof so I thought someone on this website could figure it out. What it says is: If we consider 3 non-concurrent, non parallel lines ...
0
votes
1answer
48 views

Sold two articles gaining $20$% on one and losing $20$% on the other. What is the total gain/loss?

Here is a problem,, Kethy sold two articles on one she gains $20$% and on the other $20$% loss. Find how much she gains or she lose if cost price of one is $50$% the other. Multiple choices are ...
2
votes
1answer
53 views

Solving $x + \lfloor x \rfloor = 2013x\cdot\lfloor x \rfloor + 2013^{-1}$

Find all possible solutions of the form $x = m/n$ (with $m, n$ coprime) of the equation: $$x + \lfloor x \rfloor = 2013x\cdot\lfloor x \rfloor + 2013^{-1}$$ ($\lfloor x \rfloor$ is the integer ...
4
votes
2answers
97 views

Natural numbers verifying $P(n) = n^2 - 42n + 440$, where $P(n)$ is the product of the digits

Let $P(n)$ be the product of the digits of the number $n$, with $n \in \mathbb{N}$. What is the product of all the natural numbers $n$ that verify the equation $P(n) = n^2 - 42n + 440$? I ...
-2
votes
1answer
73 views

Simplify factorials into a combinatorial formula

Is there any way to simplify this into a combinatorial formula? $$\frac{t!(n-t)!}{n!}$$
2
votes
2answers
61 views

Another way to solve this problem with complex expressions

The problem is this: Express $x$ and $y$ with $u$ and $v$, if $\dfrac{1}{x+iy} + \dfrac{1}{u+iv} = 1$ Where $x,y,u,v \in \mathbb{R}$, and $i^2 = -1$. I could solve it, but I used a hairy and ...
4
votes
2answers
245 views

Problem getting the real roots of this complex expression

I'm trying to get the real roots of this expression: $$\dfrac{1}{z-i}+\dfrac{2+i}{1+i} = \sqrt{2}$$ Where $i^2=-1$ and $z=x+iy$. I tried to simplify that with Algebra, and then separate the real ...
3
votes
3answers
130 views

Problems with trigonometry getting the power of this complex expression

I'm here because I can't finish this problem, that comes from a Russian book: Calculate $z^{40}$ where $z = \dfrac{1+i\sqrt{3}}{1-i}$ Here $i=\sqrt{-1}$. All I know right now is I need to use ...
2
votes
2answers
86 views

Drawing the region which the pedestrian can cover in 1 hour.

A straight path separates a meadow from a field. A pedestrian travels along the path at a speed of 5 km/hr, through the meadow at a speed of 4 km/hr, and through the field at a speed of 3 ...
0
votes
1answer
43 views

Multiple solutions for quadratic equation condition.

Q. All the values of m for which both roots of the equations $x^2-2mx+m^2-1=0$ are greater than −2 but less than 4, lie in the interval. Well I have done this by taking the roots,lets say, ...
3
votes
1answer
61 views

Find integral solutions for $2x^2+y^2=2\times(1007)^2+1$

Find integral solutions to the equation $$2x^2+y^2=2\times(1007)^2+1$$ I tried: I rewrote the equation as $2x^2+y^2=2028099$. I found that $y_{max}=1424$ and $y$ must be odd, so I set ...
4
votes
5answers
69 views

Pre-calculus algebra logarithm question

I don't understand how to solve this equation. Been struggling with it and don't know how to start: $$\log_2x=8+9\log_x2$$ Can someone please help me out?
0
votes
1answer
32 views

Multiplying surds

if I want to multiply $3\sqrt3$ with $3$, do I first do 3 * 3, and then $\sqrt3$ * 3? Or do I have to treat it like as one ? Very elementary question, reviewing basics that I forgot. I am looking for ...
0
votes
4answers
89 views

Factoring the Cubic Equation $2x^3 + x^2 + kx + 6$

$(x + 3)$ is a factor of $2x^3 + x^2 + kx + 6$ Find the value of $k$.
4
votes
3answers
82 views

How many solutions are there for this equation: $(x^2-x-1)^{x^2}=(x^2-x-1)$

My books says the possible solutions to $\hspace{0.2cm}$$(x^2-x-1)^{x^2}=(x^2-x-1)$ $\hspace{0.2cm}$in $\hspace{0.1cm}$$\mathbb{R}$ $\hspace{0.1cm}$ are $\hspace{0.1cm}$ $-1,1,2$ Is not ...
2
votes
2answers
163 views

How many different proofs are there that $a^n-b^n =(a-b)\sum_{i=0}^{n-1} a^i b^{n-1-i} $?

How many different proofs are there that $a^n-b^n =(a-b)\sum_{i=0}^{n-1} a^i b^{n-1-i} $ for positive integer $n$ and real $a, b$? You can use any techniques you want. My proof just uses algebra, ...
0
votes
1answer
38 views

Quadratic Equations and factorized form

I have trouble understanding why: Given the equation $x^2+4x-21=0$, the solution given is $(x+7)(x-3)=0$ in its factorization form. Using quadratic equations, $$ x = \frac{-b\pm \sqrt{b^2-4ac}}{2a} ...
1
vote
3answers
589 views

Derivation of the “Combined Work Formula”

Before I get to my question, some background: Person $A$ can paint a fence at the rate $9 \frac{hour}{fence}$ (or equivalently $\frac{1}{9} \frac{fence}{hour}$) Person $B$ can paint a fence at the ...
5
votes
2answers
117 views

What is this notation for a function? I've never seen it written like this before.

What does this mean? $$ f=\{ (x,y): y= x+2 \}$$ I don't understand what "$(x,y):$" means in regard to the problem. Also why is the $y$ inside of the $f(x)$ function. Isn't it supposed to be outside? ...
4
votes
2answers
191 views

Why we can't define $\frac{1}{0}$ to be $1$ (or anything else), but we can define $1^0$ to be $1$?

We know that we can't define division by zero "in any mathematical system that obeys the axioms of a field", because it would be inconsistent with such axioms. (1) Why can we define $a^0$ ($a\neq 0$) ...
3
votes
1answer
73 views

Geometric idea behind equations of the form $|x-a|\pm|x-b|=c$

So let's say I want to solve $$|x-a|\pm|x-b|=c$$ Using the classic multiple cases approach, one can show that the solutions are given by $$x=\frac{a+b\pm c}2 $$ But how can one make sense of this ...
0
votes
3answers
64 views

Why do the relations $ab=1/2$ and $a>b$ imply $a^2>1/2>b^2$ for positive $a,b$?

When I was reading a probstat book, I encountered an example which I am able to understand except for a formula which I am not able to grasp. It may be basic but I am not able to get it, the solution ...
1
vote
1answer
509 views

Transformation matrix from a translated-rotated coordinate system to the general coordinate system

In Figure 1, suppose $XYZ$ (in black) as my general coordinate system and $X'Y'Z'$ (orange) as another system with parallel axes respect to $XYZ$. Consider $xyz$ (green) is my 3rd coordinate system ...
2
votes
3answers
67 views

Infinite Limit Question

I am just starting limits, really stumped on this one. How do I approach this? $$\lim_{x\to -\infty} (x-2)(x-3)$$
1
vote
2answers
73 views

Area remaining after maximal number of tiles are laid on a pathway

A rectangular plot measuring $30$ m $\times$ $40$ m has a $2$ m wide pathway in the middle crosswise. Tiles of dimensions $30$ cm $\times$ $50$ cm are laid on the pathway in such a way so that no ...
1
vote
0answers
19 views

I am trying to create a growth rate that I can use for projections.

Here is our monthly growth for 2012, 2013 and 2014. I want to project monthly growth for the rest of 2014 into 2015. How can I do this? Thanks for your help! -- Math Idiotrix ...
0
votes
1answer
96 views

Finding distance using rates of change — best approach?

The question: A man drives from state $A$ to state $B$ going $60 \frac{miles}{hour}$. Then he returns from state $B$ to state $A$, driving $45 \frac{miles}{hour}$. His total driving time is $2.5 ...
0
votes
1answer
19 views

Method for finding function from graphs

I have a linear line on a graph that intersects the y axis at 0.1 and the x axis at 1. Very simple, however for the life of me I can't remember the process for finding the function expression. By ...
0
votes
1answer
85 views

Is this real number an integer?

Is this real number : $$\Big(2+\frac{10}{9}\sqrt{3}\Big)^{1/3}+\Big(2-\frac{10}{9}\sqrt{3}\Big)^{1/3}$$ an integer ? I've tried different factorization, but nothing seems to work.
2
votes
4answers
1k views

Factoring a quadratic with number in front of $x^2$

I have not yet understood how to factor a quadratic that contains a number in front of $x^2$, without using the quadratic equation. I am used to just brute forcing numbers such that AB and A + B are ...
0
votes
1answer
262 views

Factorizing Cubic equation into quadratic and linear equation?

If it is given that $x^2+x+1$ is a factor of : $ax^3+bx^2+cx+d=0$ How can we write the above equation as $\left(x^2+x+1\right)\left(ax+d\right)=0$ Where did the 'b' and 'c' dissappear.
1
vote
1answer
49 views

Why are points from this matrix geometric sequence co-planar?

Let $ M= \left[ {\begin{array}{ccc} a_{1,1} & a_{1,2} & a_{1,3} \\ a_{2,1} & a_{2,2} & a_{2,3} \\ a_{3,1} & a_{3,2} & a_{3,3} \\ \end{array} } \right] $, such ...
20
votes
5answers
2k views

Where did the negative answer come from?

The question is to evaluate $\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots }}}}$ $$x=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots }}}}$$ $$x^2=2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots }}}}$$ $$x^2=2+x$$ ...
2
votes
2answers
38 views

Im trying to find what I got wrong in simplifying

The question states " Simplify: $$\frac{6\left ( 27^{2n+3} \right )}{9^{3n+6}}$$ What I did was factoring $9$ and and $27$ to make it $3$. In the end I got a $\frac{6}{27}$ answer, simplified: ...
0
votes
1answer
28 views

Which compound interest formula would you use in this situation?

\$38,900 is used to reduce a debt of \$900,000 immediately. Then \$3,055 is paid every month. The interest rate would than be fixed for the next 4 years at the rate of 4.3% p.a. (Does not say it is ...
9
votes
6answers
5k views

How to solve the inequality $x^2>10$ using square roots?

Solve the inequality: $$x^2>10$$ How am I supposed to do this? It doesn't make sense when I take into account that if $x^2=10$ then $x=+\sqrt{10}$ and $x=-\sqrt{10}$ But how am I supposed to ...
1
vote
1answer
47 views

Proof, that equation decribes trace of curve, which is supposed to be simple

The equation, representing the trace of the curve $$ \varphi(x) = (\cos^3(t), \sin^3(t)) $$ is $1 = x^{\frac{2}{3}} + y^{\frac{2}{3}}$. Proof: Let $(x,y) = (\cos^3 t, \sin^3 t)$, then $x^{1/3} = ...