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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Problem of choice under conditions of certainty and with immediate effect.

A transport company expects to deliver 8,000 tons of goods per month for six months, 6000 tons per month for three months, 5000 tons in July, 5000 tons in August, and 4000 tons in January. Each truck ...
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1answer
47 views

Simplify factorials into a combinatorial formula

Is there any way to simplify this into a combinatorial formula? $$\frac{t!(n-t)!}{n!}$$
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2answers
52 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 ...
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2answers
222 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 ...
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3answers
101 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
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3answers
92 views

Derive $\left(\frac{1}{2}\right)^x 4^{x/2}=\frac{2^x}{2^x}=1$ [closed]

I am not able to see this, I believe that it is one of the exponent rules used but I do not see how. $\left(\frac{1}{2}\right)^x 4^{x/2}=\frac{2^x}{2^x}=1$
2
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2answers
78 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 ...
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1answer
22 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, ...
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1answer
48 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 ...
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5answers
55 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?
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1answer
23 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 ...
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0answers
65 views

The steps to becoming a Pure Mathematician

I'm a college freshmen intending to major in Mathematics. I started college with not so great grades in my calculus courses. However, I realized that to get the grades I just have to practice. I ...
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4answers
63 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 $k$.
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3answers
79 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 ...
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2answers
69 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, ...
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1answer
12 views

Quadratic Equations and factorize form

I have trouble understanding why Given the equation x^2+4x+210=0 and the solution given is (x+7)(x-3)=0 in its factorization form. Using quadratic equations x = -b\pmsqrt(b^2-4ac)/2a It works out ...
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3answers
51 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 ...
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2answers
77 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? ...
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2answers
145 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 $a^0$ ($a\neq 0$) ...
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1answer
65 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 ...
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3answers
62 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 ...
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1answer
29 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 ...
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3answers
56 views

Infinite Limit Question

I am just starting limits, really stumped on this one. How do I approach this? $$\lim_{x\to -\infty} (x-2)(x-3)$$
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2answers
56 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 ...
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0answers
15 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 ...
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1answer
31 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 ...
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1answer
12 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 ...
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1answer
80 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.
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4answers
45 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 ...
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1answer
42 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.
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1answer
39 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 ...
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5answers
1k views

Where did the negative answer come from?

The question is to evaluate $\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots }}}}$ $$x=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots }}}}$$ $$x^2=2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots }}}}$$ $$x^2=2+x$$ ...
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1answer
43 views

Problem involving Basic Trigonometry [closed]

If $\sqrt{2} \cos{A}=\cos{B} +\cos^3{B}$, $\sqrt{2} \sin{A}=\sin{B} -\sin^3{B}$, then find the value of $\sin(A-B)$.
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2answers
34 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
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1answer
20 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 ...
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6answers
554 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 ...
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1answer
20 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} = ...
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2answers
51 views

Solve for positive reals

Solve for positive reals $x,y$ $(x+y)(1+\frac{1}{xy})+4=2(\sqrt{2x+1}+\sqrt{2y+1})$ I started by accumulating the terms of $x$ and then used AM-GM inequality but unsuccessfully....
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3answers
112 views

Mathematical way to solve integer numbers $217 = (20x+3)r+x$

Is there any mathematical way to find the integer numbers that solve the following equation: $$217 = (20x+3)r+x$$
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1answer
41 views

Find polynomials $f (x)$, $g(x)$, and $h(x)$

In an elementary Algebra book (101 problems in Algebra) there was a question I solved but when I looked at the solutions I didn't get it. it says find Polynomials $f(x)$, $g(x)$, $h(x)$ such that for ...
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1answer
34 views

Domain and Range of relations

I am having problems understanding how to solve/find the domain and range for these 2 equations. 1) $y-3x^2 = 2$ 2) ${(x,y); x=3}$ Any help would be much appreciated, Thanks!
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3answers
45 views

Is there anyway to find the values for equation with two variables

Is there mathematical anyway to find the values for equation with two variables? as example: I have y = (20x+3)r+x and y is 50. then what is x and r values?
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2answers
46 views

How can I find the R values

I have the following equations : $$\begin{cases}K = \frac{B – 3}{20}\\ K = (20S+3)R+S\\ K = 20S^2 + (20N+7)S + N\\ N= R - S \end{cases}$$ - And I have the $B$ values, e.g : 834343, 3253538, ...
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3answers
30 views

Calculating the angle for a path between two nodes in a graph

I want to (programatically) draw an edge between two nodes in a graph, starting on the outside of the nodes. Below is an illustration of what I'm (poorly) trying to describe: I have the $(x,y)$ ...
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1answer
39 views

Is there a name for these binary operations?

In computer programming, I often encounter the need to give the binary operations: The greatest multiple of $y$ that is not greater than $x$ $41 \circ_1 6 = 36$ $3.2 \circ_1 0.5 = 3$ The least ...
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3answers
32 views

Calculating the selling price of a house, given the mortgage, commission, and closing costs

I'm trying to help my wife study for her real estate exam, I thought I was good at math (although I haven't done any in years). Here is the question at hand : You sell a home for a client, and ...
2
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2answers
84 views

Any idea how to linearize this equation? $X^2-Y^2=aZ+bZ^2$

The intention is to linearize this equation $X^2-Y^2=aZ+bZ^2$ into something which looks like $Z=mX+nY+c$ so that a graph of $Z$ against $X$ or $Y$ can be plotted. X,Y,Z are variables while a,b,c are ...
0
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0answers
38 views

Easy leg and heads problem

Some chickens and rabbits have a total of 100 feet. If each chicken is exchanged for a rabbit and each rabbit is exchanged for a chicken there would be a total of 86 feet. How many chickens are there? ...
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2answers
31 views

Sketching graphs abs value functions

how do I go forward with sketching the graphs of the following two functions? i) $y(t)=|2+t^3|$ ii) $f(x)=4x+|4x-1|$ thanks in advance!
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5answers
66 views

Solving limit given a limit

Given the fact that $$\lim \frac{\sin(h)}{h}=1,\ \mbox{as}\ h\to0$$ compute the following limit: $$\lim\frac{\sin(x+h) - \sin(x)}{h}\ \mbox{as}\ h\to0.$$ How would I go about solving this problem? ...