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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1
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3answers
91 views

Finding time needed for a task, given two person's distinct task/time rates

John can dig constantly at $15$ inches per minute and Linda can dig constantly at $45$ inches per minute. A certain hole can be dug by John in $12$ hours. If hole is dug by John for half the time ...
0
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1answer
342 views

Maximize area of a rectangle

A rectangular lot is bordered on one side by a building and the other 3 sides by $800$m of fencing. Determine the area of the largest lot possible.
0
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1answer
135 views

Converting equation to $y = mx + b$

My sister have an assignment of converting below equation to slope as $y = mx + b$ $xy = 4$ Can anyone help? thanks in advance. ^_^
1
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1answer
32 views

Finding a concave function that minimize the middle value while the boundary values are fixed

This question came to me while I was listening to Dominik's talk this afternoon. First, let me remind you what does f is concave mean. It means f satisfies $pf(x)+(1-p)f(y)\le f(px+(1-p)y)$, $\forall ...
1
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1answer
38 views

Finding % of remaining students in the class would be girls?

In class of $120$ students, boys constitute $40$% of total. If $\dfrac 13^{rd}$ of boys and $4$ girls drop out of class to join a camp , what % of remaining students in the class would be girls ?
2
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1answer
25 views

Linear Systems: Application

Zoltan(who is behind Mali) and Mali are 18 km apart and begin to walk at the same time. If they walk in the same direction they meet after 6h. If they walk towards each other they meet in 2h. Find ...
1
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1answer
295 views

Linear Transformation of a straight line

Let $L_{1}: x-y-2=0$ be a straight line in the x-y coordinate system. Find a coordinate system $(x_{1},y_{1})$ having its origin at $(0,0)$ and relative to which $L_{1}$ has equation $y_{1}= ...
4
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4answers
440 views

Find the values of the constants in the following identity $2x^3+3x^2-14x-5=(ax+b)(x+3)(x+1)+C$

I'm working through identities but I can't figure out how to get further than multiplying out the above to get : $$2x^3+3x^2-14x-5=2ax^3+3ax^2+3ax+bx^2+3bx+bx+3b+C$$ can someone give me a hint on ...
2
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2answers
147 views

Michelson-Morley Experiment

I've looked everywhere and I cannot find a complete derivation that includes the step I'm looking for lol...hopefully this will add another link for google. So the full time for light to travel both ...
4
votes
1answer
554 views

Concrete Mathematics Prerequisite Question

I've been very interested in the book Concrete Mathematics (Graham,Knuth,Patashnik) and I've been reading it for the past few weeks. I'm at the chapter about Sums (Chapter 2), specificaly, the lesson ...
1
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1answer
912 views

Normalizing an exponential function

Given the equation $a^\frac yx + a^x=b$ is there a way to normalize this function into a form where $y=$...? In short can I express $y$ in terms of $x$ if $a$ and $b$ are constants?
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9answers
3k views

What is sum of all positive odd integers less than $1000$?

If the sum of all positive even integers less than $1000$ is $ A $ , what is the sum of all positive odd integers less than $1000$?
0
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2answers
59 views

How do I find all real numbers of Y with a nontrivial solution?

My problem is: Find all real values of Y, if any, for which the system has a nontrivial solution. $$2X_1 + 3X_2 = YX_1$$ $$4X_1 + 3X_2 = YX_2$$ Thank you.
2
votes
4answers
85 views

Finding the ratio of two persons time spent driving to a meeting

Mark and pat drive separately to a meeting. mark's average driving speed is $1/3rd$ greater than pat's and mark drives twice as many miles as pat. What is ratio of number of hours mark spends driving ...
7
votes
0answers
221 views

Proving a geometric inequality without Lagrange multipliers

Let $e=(1,1,\ldots,1)$ be the $n$-dimensional vector consisting only of ones. Let $r=\sqrt{\dfrac{n}{n-1}}$ and $\alpha \in (0,1)$ fixed. Given a vector $x=(x_1,x_2,\ldots,x_n) \in \mathbb R^n$ such ...
1
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3answers
670 views

Given $y=\arccos(x)$ find $\arcsin(x)$ in terms of y

Given that $y = \arccos x$, $ - 1 \le x \le 1\,and\,0 \le y \le \pi $, express $\arcsin x$ in terms of y. The best I know how to do this is is: $$\eqalign{ & \cos y = x \cr & ...
5
votes
2answers
704 views

Every perfect cube is the difference of two perfect squares?

How would you prove this without induction? I know that one easy way is using Al Kharchi's principle (namely that $1^3+2^3+3^3+...+n^3=(1+2+3+...+n)^2$), but are there other ways? Thanks!
2
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2answers
60 views

Converting data to a specified range

I am trying to convert data to a particular range of 0-10. Actual Data may vary from - 50000 - 26214400. I have broken this down in to 4 parts as follows - ...
0
votes
1answer
30 views

algorithm to assign points to winning users

I have a set of users who have won a game ('jim', 12), ('james', 54), ('john', 76), ('dave', 22), ('garry', 34), ('stuart', 16) I would to award them a share of ...
2
votes
3answers
115 views

Show that $x=2\ln(3x-2)$ can be written as $x=\frac{1}{3}(e^{x/2}+2)$

Show that $x=2\ln(3x-2)$ can be written as $x=\dfrac{1}{3}(e^{x/2}+2)$. Is there a rule for this?
2
votes
2answers
90 views

Why differentiating a function is similar to differentiating its log function?

$f(6;p)=\binom{25}{6}p^6(1-p)^6,\quad 0\le p\le1$ I have three questions: $(1)$To find the relative maxima of p, the process is to take the derivative of the function with respect to $p$ equal to ...
0
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3answers
973 views

One-to-one and onto functions

Let $\mathbb{N}$ be the set of all positive integers. a). Define a function $f: \mathbb{N} \rightarrow \mathbb{N}$ that is one-to-one but not onto. b). Define a function $g: \mathbb{N} \rightarrow ...
3
votes
2answers
151 views

What is the defining characteristic of a quadratic function?

I'm helping a high school student prepare for an exam, and I'm unsure how to answer this... Why is $x^3+2x^2$ not quadratic? I thought anything that had a power of 2 was quadratic.
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1answer
541 views

finding the coordinates of a point of intersection: 3d sphere and plane

How to find the coordinates of one point on the interaction of the sphere $$(x-1)^2+(y-2)^2+(z-4)^2= 25$$ and the plane $z=4$. I was trying to solve this I got it down to $x+y=8$ but then when I ...
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3answers
77 views

Finding unknowns in terms of polynomials

Consider the following equation $$(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2})(y_{1}^{2}+y_{2}^{2}+y_{3}^{2}+y_{4}^{2}+y_{5}^{2}) = z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2}+z_{5}^{2}$$ How ...
0
votes
2answers
38 views

Find values of the constants in the following identity: x^4+Ax^3 + 5x^2 + x + 3 = (x^2+4)(x^2-x+B)+Cx+ D

Another question on identities: $$x^4+Ax^3 + 5x^2 + x + 3 = (x^2+4)(x^2-x+B)+Cx+ D$$ How can I find the coefficients for this? I've got as far as multiplying out the brackets to get: $$x^4+Ax^3 + ...
13
votes
3answers
521 views

Is there any “superlogarithm” or something to solve $x^x$? [duplicate]

Is there any "superlogarithm" or something to solve an equation like this: $$x^x = 10?$$
0
votes
2answers
27 views

Find the values of the constants in the following identitity $x^4+4/x^4 = (x^2-A/X^2)^2+B$

A step by step solution would be preferred for the following question : Find the values of the constants in the following identitity $x^4+4/x^4 = (x^2-A/X^2)^2+B$. so far I managed to substitute $x$ ...
0
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1answer
22 views

A formula to find the organs's value from $1$ to $100$.

We have a variable named NUMBER This variable, can hold ANY number from $1$ to $100$. Let's mark that number as $X$. We know that: $x(1) = 1000$ Coins $x(100) = 250$ Coins I need to write down a ...
1
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4answers
385 views

how to know of the number of real roots?

Let $ax^4 +bx^3 +cx^3 +dx + e = 0$ with $a,b,c,d,e\in\mathbb R$. I would like to know, how can I determine the condition for the polynomial to have exactly three distinct real solutions. one has to be ...
0
votes
3answers
100 views

Find the values of the constants in the following identities $A(x^2-1)+B(x-1)+c = (3x-1)(x+1)$

I'm stuck on a basic question regarding identities. $A(x^2-1)+B(x-1)+C = (3x-1)(x+1)$ I've managed to substitute $x$ for $1$ to work out C is $4$. However, I'm unsure how to work out A and B ...
51
votes
6answers
4k views

Continued fraction fallacy: $1=2$

It's easy to check that for any natural $n$ $$\frac{n+1}{n}=\cfrac{1}{2-\cfrac{n+2}{n+1}}.$$ Now, ...
2
votes
1answer
35 views

Solving a set of two polynomial equations

For a given $a,b,c,d$ in $\mathbb{R}$, I want to prove that if $$ac-bd=0 \quad \text{and} \quad ad+bc=0$$ then $a=b=0$ or $c=d=0$. I am able to prove this in a long and cumbersome way, and I'm ...
2
votes
3answers
162 views

How find $a$ such that $x^2-\sqrt{a-x}=a$ has exactly two real solutions

Consider the equation $$ x^2-\sqrt{a-x}=a.$$ I wish to determine the values of $a$ for which the above equation has exactly two real solutions (for $x$). My idea: ...
1
vote
2answers
68 views

System of linear equation problem

A basketball club offers two types of memberships. Membership Y: No registration fee and $\$15$ per hour for court time. Membership Z: $\$40$ registration fee and $\$5$ per hour for court time. ...
4
votes
4answers
1k views

How to find out X in a trinomial

How can I find out what X equals in this? $$x^2 - 2x - 3 = 117$$ How would I get started? I'm truly stuck.
3
votes
2answers
450 views

The system of equations $x^2 + y^2 - x - 2y = 0$ and $x + 2y = c$

I have $(1.) \quad x^2 + y^2 - x - 2y = 0 \\ (2.) \quad x + 2y = c$ Solving for $y$ in $(2.)$ gives $y = (c - x) / 2$ Is there a way to simplify equation $(1.)$? Because at the end I arrive at ...
2
votes
1answer
190 views

Complex numbers true or false

Are there any complex numbers "z" that satisfy this equation? $$z=-\bar z?$$
1
vote
3answers
116 views

Is there a way to factor $uv-u-v-1$?

$uv-u-v-1$. I tried $(u+1)(v-1)$ and it's almost correct but I can't quite get it/ This is part of a limit problem I'm doing. Thanks!
3
votes
4answers
440 views

How $\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2 = \frac{\sum_{i=1}^n (X_i - \bar X)^2}{n}$

How $\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2 = \frac{\sum_{i=1}^n (X_i - \bar X)^2}{n}$ i have tried to do that by the following procedure: $\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2$ ...
0
votes
1answer
128 views

Simple inequality with unknown in the exponent

Let $0<\alpha\ll1$ I have the following inequality: $$ 2\alpha^2x\geq \alpha^{2x} $$ It looks trivial, but I wasn't able to find the $x$ that verify the condition. Anyone any clue?
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1answer
83 views

Solve this system of equations

Solve the system of equations: $$\begin{cases}\dfrac{\sqrt{x}}{2}+\dfrac{y+1}{\sqrt{x}}=\dfrac{x+y}{x}\\[0.1in] 2\sqrt{y}+\dfrac{x-2}{\sqrt{y}}=\dfrac{2y+x}{y}\end{cases}$$
0
votes
1answer
97 views

Partial fractions on complex function for Laurent series

Whats the proper way to calculate the partial fractions for functions like these $$ f(z)=\frac{z}{z^2(z+1)} \\ f(z)=\frac{z}{(2-z)^3z} $$ before calculating the Laurent series?
0
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1answer
119 views

Find vector and parametric equations of span v

How do I find vector and parametric equations of span $\{v\}$? Need help on steps to take or description.
5
votes
1answer
124 views

Conceptual questions about cross product rules

I am studying cross products and I just cannot get my head around the materials. Can someone explain to me why... a) The cross product $\vec{a}\times \vec{b}$ is normal to $\vec{a}$ and $\vec{b}$. ...
1
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6answers
80 views

Product-to-sum formulas?

My old pre-calculus book says: $$\sin u\cos v=\frac{1}{2}[\sin (u+v)+\sin(u-v)]$$ and $$\cos u \sin v=\frac{1}{2}[\sin(u+v)-\sin(u-v)]$$ I don't understand why there is a difference, since ...
2
votes
4answers
91 views

describe the domain of a function $f(x, y)$

Describe the domain of the function $f(x,y) = \ln(7 - x - y)$. I have the answer narrowed down but I am not sure if it would be $\{(x,y) \mid y ≤ -x + 7\}$ or $\{(x,y) \mid y < -x + 7\}$ please ...
4
votes
6answers
245 views

How to simplify $\frac{1-\frac{1-x}{1-2x}}{1-2\frac{1-x}{1-2x}}$?

$$\frac{1-\frac{1-x}{1-2x}}{1-2\frac{1-x}{1-2x}}$$ I have been staring at it for ages and know that it simplifies to $x$, but have been unable to make any significant progress. I have tried doing ...
2
votes
2answers
67 views

Given $\frac {a\cdot y}{b\cdot x} = \frac CD$, find $y$.

That's a pretty easy one... I have the following equality : $\dfrac {a\cdot y}{b\cdot x} = \dfrac CD$ and I want to leave $y$ alone so I move "$b\cdot x$" to the other side $$a\cdot y= \dfrac ...
5
votes
6answers
362 views

Is my proof correct? $2^n=x^2+23$ has an infinite number of (integer) solutions.

This is how I tried to prove it. Is it correct? Thanks!! $2^n = x^2+23$ $x^2$ must be odd, therefore $x^2 = 4k+1$, where $k \in \mathbb{N}$. $2^n=4k+24$ $k=2(2^{n-3}-3)$ Since $x^2=4k+1$,$ \ \ \ ...