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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2
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0answers
54 views

Find the roots of the equation

How many roots does the equation $$x^{x^x}=(x^x)^x\\$$ have in $\\\mathbb{R}$? My observations:I observed that $x=-1,1,2$ are its roots. Are there other roots of this equation?And how we can find ...
0
votes
0answers
15 views

Locus given by a pair of scissors sliding along the ground.

I came up with this problem when dragging a pair of scissors along the ground. The question is, more mathematically: Suppose there is a point (a,0) and a point (0,b) with a fixed distance m between ...
1
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3answers
24 views

Use a function to represent positive real numbers?

Is it correct to define the positive real numbers as $\{f(x) = x^2\mid x \in \mathbb R\}$?
3
votes
3answers
237 views

Show that inequality holds

How would you show that the following inequality holds? Could you please write your reasoning by solving this problem too? $a^2 + b^2 + c^2 \ge ab + bc + ca$ for all positive integers a, b, c I ...
0
votes
0answers
31 views

When Is This Equation an Integer?

I'm working on a math problem, but I've gotten stuck. When does $$-\frac{4(2^Q-3^Q)}{2^{Q+S}-2*3^Q}$$ produce an integer?
1
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1answer
14 views

Determine values of the constants a & b so the limit exists and is equal to f(2) in a piece-wise function

I am trying to determine the values of the constants a & b in a piece-wise function that has to satisfy these parameters: The limit f(x) as x approaches 2 does exist and is equal to f(2) The ...
-4
votes
1answer
35 views

please help me with this [on hold]

the cooling system of a car has a capacity of 15 liters if the system is currently filled with a mixture that is 10% antifreez how much of this mixture should be drained and replaced with pure ...
1
vote
2answers
19 views

How is it sometimes helpful to use cross multiplication in order to complete proportions with a variable?

How can it be helpful to do cross multiplication with proportions with variables such as ${2\over 4}={3\over x}$? In this one, the value of x has to be found. It can be found this way: 1. Do the ...
1
vote
4answers
27 views

systems of linear equations intuition

I want to know why in a system of linear equations I'm allowed to sum or subtract the equations. I can't get the intuition of why I can do that to solve for the equations.
5
votes
7answers
446 views

Is the number 0.2343434343434.. rational? [duplicate]

Consider the following number: $$x=0.23434343434\dots$$ My question is whether this number is rational or irrational, and how can I make sure that a specific number is rational if it was written in ...
1
vote
1answer
19 views

Simplification of $|a+b|^2$ for $a,b \in \mathbb{C}$

How do I simplify $|a+b|^2$, where $a,b \in \mathbb{C}$ and $|a|=|b|=1$? I know that the result is $4-|a-b|^2$, but I would like to be be explained how to do the simplification in the most elegant ...
1
vote
2answers
548 views

Real world situation with System of Equation with 3 variables?

Where do you run into a real world situation involving 3 variables and 3 equations? Can someone think of a specific example from business, etc? I recall taking an operations research course that ...
0
votes
1answer
16 views

Deal with non standard form of conic

I want to know how can I calculate latus rectum, tangent at vertex, vertex and axes of a parabola whose equation is not standard. For example, the parabola: $$ 4x^2 - 4xy + y^2 - 10 y - 19 = 0 $$
0
votes
4answers
24 views

Help in proving an algebraic identity involving powers of binomials.

For some reason I found this equation: $(1 + x)^n - 1 = x \sum\limits_{k=0}^{n-1} (1+x)^k$ I think that this is an identity. If for instance one expands the powers and the sum for n = 4, the ...
5
votes
8answers
348 views

If $\,\,x+\dfrac{1}{x}=5,\,\,$ find $\,\,x^5+\dfrac{1}{x^5}$.

If $x>0$ and $\,x+\dfrac{1}{x}=5,\,$ find $\,x^5+\dfrac{1}{x^5}$. Is there any other way find it? $$ \left(x^2+\frac{1}{x^2}\right)\left(x^3+\frac{1}{x^3}\right)=23\cdot 110. $$ Thanks
0
votes
2answers
27 views

Asymptotic approximation of the arctangent?

That is, I am looking for an algebraic function $f(x)$ that approximates $\arctan x$ for large values of $x$. The approximation could be reasonably modest -- perhaps something like $$\tan (f(x)) = ...
1
vote
1answer
56 views

Triplets of distinct integers > 1 that return integer values.

If $(A, B, C)$ are distinct integers $> 1$, and $$f(A, B, C) = \frac{\frac{A^2-1}{A} + \frac{B^2-1}{B}}{\frac{C^2-1}{C}},$$ then for what (if any) triplets $(A, B, C)$ is $f(A, B, C)$ an integer? ...
1
vote
1answer
972 views

How can convert the general form of ellipse equation in the standard form?

How can convert the general form of ellipse equation in the standard form? $$-x+2y+x^2+xy+y^2=0$$ Thank you in advance?
0
votes
1answer
11 views

Contradiction - Equivalence of polynomials

I think I'm having a brain fart. Please tell me if my reasoning is correct. Suppose you have a polynomial-function $f(x)$ of degree $N$ that has coefficients $a_{0 \leq j \leq N}$ and roots $r_{0 ...
2
votes
2answers
237 views

What do these extra solutions mean?

I'm trying to find a constant $a$ such that $n(n+1)(n+2)(n+3)$ is equivalent to $(n^2+an)(n^2+an+2)$. Clearly by inspection, we have $a=3$. However, say I wish to substitute $n=-1$. We get our ...
0
votes
2answers
98 views

How to solve $100x^{99} + \cos x = 0$?

I want to know how to solve such equations without a graphing calculator, so please show the steps. $$100 x^{99} + \cos x = 0$$
1
vote
2answers
877 views

Solving system of equations with R

Suppose I have the following function : $f(x,y,z,a)= \cos(ax) + 12y^2 - 9az$ and I want to solve the following syste of equations. $ f(x,y,z,1)= 10 $, $f(x,y,z,5)= 7 $, and $f(x,y,z,-3)= 17 ...
-3
votes
1answer
45 views

Ladder against a wall.

Having a bit of a problem with a question. There is a 4m ladder leaving against a wall. There is a box in between The ladder and wall. The box is a cubic metre. I have found a quartic to find the ...
0
votes
1answer
8 views

Compare sales growth

So I’m trying to measure the sales growth of specific salesman. Any salesman has the highest sales growth (min $4\%$) will receive a giftcard. However I found it not fair to compare when a saleman ...
2
votes
2answers
45 views

Cool little system of equations.

Solving the system of equations for integers: $2^a+3^b=5^b$ $3^a+6^b=9^b$ How is it done? I tried substituting the $2^a$ from the first equation into the second, and dividing the two equations by ...
0
votes
0answers
11 views

Polynomial Long Division with Divisor<Dividend

So here's the problem... 20x^3-4/5x^2-3 When I divide this I get 20x^3 -4 -20x^3 +12x 12x-4/5x^2-3 So 5x^2 goes into 12x how many times? It doesn't seem to. So how do I solve this?
1
vote
0answers
24 views

Question about series and how the pattern idea works

Two Questions: When you are given: $1, 2, 3, .... , n$ How do you know that in the $...$ that it continues the $x_{n-1} + 1$ pattern? Is it the definition of series? Secondly: Do partial sums ...
0
votes
2answers
33 views

Solve for x, y and z

$$x+y-2z=5 (1)\\ x +z=4 (2)\\ -z=6 (3)$$ $$2\cdot(2): 2x+2z=8 (4)\\ (1)+(4): 3x+y =13 (A)$$ $$(2)+(3): x=10 (B)$$ $$(A)-(B)= 3+y=3\\ y=3-3\\ y=0$$ Substitute $y$ into $(A)$ ...
0
votes
0answers
18 views

Prove $\log_ab+\log_bc+\log_ca\geq1+\log_{ab}bc+\log_{bc}ab$

Prove inequality $$\log_ab+\log_bc+\log_ca\geq1+\log_{ab}bc+\log_{bc}ab$$ for $a>1,b>1,c>1.$ We noted $x=\lg a,y=\lg b, z=\lg c $ and wrote inequality in the form ...
2
votes
1answer
13k views

Finding Revenue Function and Max Revenue

Studying for a midterm. The demand function for a manufacture's product is $p=1000-\frac1{80} q$ Where $p$ is the price (in dollars) per unit when $q$ units are demanded (per week) by consumers. ...
6
votes
2answers
574 views

Beautiful problem on a progression

$\{x_n\}$ is a sequence defined as follows: $x_1=20,\quad x_2=14,\quad x_{n+2}=x_n - \frac{1}{x_{n+1}}$. Prove that $0$ is among the members of this sequence. Find its number. I tried some stuff ...
0
votes
2answers
30 views

Expressing $\frac{1}{4n^2-1}$ as a partial fraction

I was asked to express $$\frac{1}{4n^2-1}$$ as a partial fraction. I have no clue as to what I should break this into. For example I know : $$\frac{1}{n(n-1)}= \frac {A}{n} + \frac {B}{n-1}$$ ...
0
votes
2answers
86 views

Quicker way to compare numbers without calculator

Question: Find the order of $(1/2)^{1/2}$, $(1/e)^{1/e}$, $(1/3)^{1/4}$ without using calculator. Extra constraint: You only have about 150 seconds to do it, failing to do so will eh... make you run ...
3
votes
1answer
42 views

Arithmetic progression with common difference 2061

If there are 30 consequent members of an arithmetic progression with CD of 2061, show that among them are at most 20 squares of natural numbers. I wrote out $a_1$ through $a_{30}$ and tried to find ...
0
votes
1answer
28 views

Expand $(\frac{x}{3}+\frac{x^2}{4})^2$

$(\frac{x}{3}+\frac{x^2}{4})^2$ I know that the special product of $(a+b)^2$ is $a^2+2ab+b^2$ they said the answer is $\frac{x^4}{16}+\frac{x^3}{6}+\frac{x^2}{9}$ I don't understand how they got ...
1
vote
2answers
57 views

$ay^3 + xy = ab^3$, can I isolate $y$?

I was wondering how much force it would take to compress a sphere of air (assuming Boyle's Law instead of the Real Gas laws, ignoring the engineering method of applying said force), so I started with ...
3
votes
1answer
29 views

Prelude to Cauchy-Schwarz, Quadratic proof.

I have a problem in trying to prove the following observation: "Show that if $ a,b,c \in \mathbb{R} $ are such that for all $ \lambda \in \mathbb{R} $, $a\lambda^2 + b\lambda +c \geq 0 $ then $ b^2 - ...
1
vote
5answers
34 views

rational function limit involving factorials

I posted something similar but someone edited the question and added the wrong equation, which gave irrelevant responses. Lim (2n-1)!/(2n)^n as n approach infinity. Any method, I would just like a ...
4
votes
4answers
45 views

How to solve a convoluted absolute value inequality?

$$ \lvert \lvert x-2\rvert -3\rvert \lt 5 $$ How can I attack this the best way? I see that both sides are positive. Squaring yields: $$ \lvert x-2\rvert ^2 -6 \lvert x-2\rvert +9\lt 25 $$ $$ ...
0
votes
3answers
91 views

How do you factor this using complete the square? $6+12y-36y^2$

I'm so embarrassed that I'm stuck on this simple algebra problem that is embedded in an integral, but I honestly don't understand how this is factored into $a^2-u^2$ Here are my exact steps: ...
-1
votes
1answer
26 views

How to compute inequality that involves logarithm

So I was reading a math book and I faced with expression I could not solve. Well, I even do not know how to begin, really. I understand that in order to compute power we need to find a logarithm. ...
-1
votes
2answers
19 views

Prove that the image of $f: (0, \infty) \to R$ is contained in $[2, \infty)$. [on hold]

where $f(x) = x + 1/x$ Any help is appreciated, what I did was completely wrong haha..
0
votes
1answer
25 views

simplify using the difference quotient when $f(x)=2/x^2$

I am trying to simplify a difference quotient with the form $$\frac{f(x+h)-f(h)}/{h}$$ if $f(x)=2/x^2$ I have attempted to cancel out the denominator of the numerator by the least common denominator ...
0
votes
2answers
26 views

Factorial simplification rules

I want to know if the following simplification that i did holds true $$(2(n+1))! = 2(n+1)! = 2(n+1)(2n)!$$ and if not true what other simplification can work for it , it's a question about series ...
0
votes
1answer
45 views

Abstract algebra

Assuming there is a real number $x$ with $ x^3 =7$, prove that $x$ is irrational. I started the proof by contradiction, and I got to the point that $7q^3 = p^3$, but I don't know what should I do ...
6
votes
3answers
87 views

Why is Division harder than Multiplication?

Both conceptually and computationally it feels easier to see that: $ 6 \cdot 3.7 = 22.2$ than it is to see that $ 22.2 \div 6 = 3.7 $. Thoughts about the roots of this asymmetry? An analogous ...
2
votes
1answer
28 views

I need help solving $3e^{2x}-1=\frac{1}{2}$

I am trying to solve $3e^{2x}-1=\frac{1}{2}$. Here is my work: $3e^{2x}-1=\frac{1}{2}$ $3e^{2x} =\frac{1}{2}+1$ $e^{2x} =\frac{1.5}{3}$ $\ln{e^{2x}} =\ln{(\frac{1}{6})}$ $2x ...
2
votes
2answers
16 views

Simplify a limit problem with the difference quotient $(g(x)-g(a))/(x-a)$ given $g(x) = -3x^2+8x+12$

I am trying to simplify this limit problem using the difference quotient, but I am not sure how to cancel out the a,s since it is g(x) and not g(x+a). Here is my work so far: $$ ...
0
votes
0answers
43 views

Help in writing a nasty expression in nice closed form

This question is abouting re-writing a product in nice closed form. I have the following $$f(v_1) = \left(\sum_{i=1}^K \pi \lambda_i \delta_1 v_1^{\delta_1-1} P_i^{\delta_1} e^{-\beta_i ...
3
votes
3answers
22 views

Calculus 1: Find the limit as x approaches 4 of $\frac{3-\sqrt{x+5}}{x-4}$

I understand how to find limits, but for some reason I cannot figure out the algebra of this problem. I tried multiplying by the conjugate and end up with 0/0. When I check on my calculator, or apply ...