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.

learn more… | top users | synonyms (2)

-4
votes
1answer
34 views

If a and b are the two solutions to x^2 - x - 2=0, then a+b=? [on hold]

If a and b are the two solutions to x^2 - x - 2=0, then a+b= ? A. -1 B. 0 C. 1 D. 3 E. 5
2
votes
0answers
30 views

What is the (currently) optimal root finding algorithm for multivariate functions? [on hold]

Let's say we wish to find the roots of the function: $f(x,y,\cdots) = 0 \;,$ so, for a minimal example: $xy - 1 = 0 \; .$ I know there are different methods to solve this problem for the ...
0
votes
1answer
16 views

$P$ is a point on a hyperbola whose focal points are $F_1$ and $F_2$. $Q$ on the line that bisects $\angle F_1PF_2$. Prove $|PF_1-PF_2|>|QF_1-QF_2|$.

$\require{cancel}$ Sorry for the grammatical mistake in the title; it was needed to keep the title under 150 characters. $P$ is a point on a hyperbola whose focal points are $F_1$ and $F_2$. $Q$ is ...
1
vote
1answer
26 views

Quotient set cardinal in $\mathbb{Z}_{12}$

In $\mathbb{Z}_{12}$ define the equivalence relation xRy if $x^2 = y^2$ Then what is the cardinal of the quotient set?
0
votes
2answers
37 views

Largest number of pairs that can be added while keeping the population at least 60% male

I'm doing problems from the AoPS Algebra Beginner's book. There's this problem that states the following, At her ranch, Georgia starts an animal shelter to save dogs. After the first three days, she ...
1
vote
2answers
102 views

Can one use logarithms to solve the equations $2=3^x + x$ and $2=3^x x$?

Could someone explain how would you solve: $$2=3^x + x$$ and $$2=3^x \cdot x$$ I can only solve halfway through. And why is $$10^{\log (x)}= x$$ Thanks
1
vote
2answers
62 views

How to find the domain and range of $f(x) = \sqrt{x^2-2x+5}$?

This is the function: $$f(x) = \sqrt{x^2-2x+5}$$ Edit: normally what I would do is this: Since it's a square root function, the thing inside the root has to be $\ge 0$. So, $(x^2 - 2x+5)\ge 0$. Then ...
0
votes
1answer
25 views

Standard form of trig equations

The standard form for any trig equation is y=Asin(B(x-D))+C (I'm just using sine in the equation). For the "D" which is the horizontal translation, if D is added does the graph move left or right, ...
1
vote
1answer
32 views

Volume of a ellipsoidal shape

I was given the following question: My approach so far was to create a parabolic function: y = 25/2 - (25^2)/392 Then I integrate from x = 0 to x = 14 Volume = 2 * pi * integral of y ^ 2 The ...
-1
votes
2answers
24 views

Graph of $\log_2(2-x)$: what is wrong in my transformational approach?

In the graph of $\log_2(2-x)$ can I have the transformational approach of $\log_2 x$ >> $\log_2(-x)$ >> $\log_2(-x+2)$ or $\log_2(2-x)$ but after all this graph comes wrong but with differential ...
0
votes
3answers
51 views

Factor the expression completely.

$$(a^{ 2 }+1)^{ 2 }-7(a^{ 2 }+1)+10$$ So far I got: $$(a^{ 2 }+1)(a^2+1)-7a^{ 2 }+3$$ I feel like I am going about this the wrong way. I need a push in the right direction.
1
vote
2answers
19 views

Graphing the secant function, $y=2\sec 2\theta$

I am asked to graph the following equation $y = 2 \sec {2\theta}$. Since the equation just has a $ 2\theta$ after the secant, is it correct to say that there is no phase shift? If I would start ...
2
votes
1answer
47 views

Is it possible to accurately calculate an irregularly shaped frustum's volume?

I have the following water basin Now imagine this basin is filled with water to the top, is there anyway to accurately calculate the volume of water stored in it using only top and bottom areas A1 ...
0
votes
5answers
52 views

inequality with absolute value?

Solve the given inequality by interpreting it as a statement about distance on the real line: $$|x+1| \gt|x-3|$$ anyone know how to go about this problem?
0
votes
0answers
41 views

Can the inequality $a^3 + b^3 + c^3 \ge a^2b + ac^3 + b^2c$ be derived from arithmetic-geometric means? [duplicate]

The inequality goes as follow: $$a^3 + b^3 + c^3 \ge a^2b + ac^3 + b^2c$$ Where $a,b,$ and $c$ are positive real numbers. Also, can it be solved using am-gm?
0
votes
1answer
25 views

Cardinal of the quotient set

Let $X = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$. In $X \times X$ define the equivalence $(a,b)\:\mathcal{R}\:(c,d)$ if $a+b=c+d$. Then what is the cardinal of the quotient set? I know that $|X| = 9$, so $| X ...
1
vote
3answers
50 views

Simple use of log

I am struggling to see how we can go from the first expression to the second: $$\begin{align} 2\log_3 12 - 4\log_3 6 &= \log_3 \left ( \frac{4^2 \cdot 3^2}{2^4 \cdot 3^4} \right )\\ &= \log_3 ...
1
vote
1answer
33 views

What is the value of the mean of these numbers?

Given that $13^{a+b}=13^{xy}=13^{13}$, what is the mean of $a, b, x$ and $y$? What I tried: the mean is $\frac{a+b+x+y}{4}$. One can infer that $a+b=13$ so that the mean is $\frac{13+x+y}{4}$. I ...
1
vote
4answers
54 views

Converting repeating decimal to fraction

How do I conver $0.297$ to a fraction, if the 2 and 9 are repeating? The non repeating number is in the middle, so I am not sure how to proceed from here. Any help is appreciated
1
vote
1answer
40 views

Formula to find the value after taking the square root of a number $n$ times?

How can I find the value after taking the square root of a number $n$ times? For example: $\sqrt{a}$, $\sqrt{\sqrt{a}}$, $\sqrt{\sqrt{\sqrt{a}}}$, $\sqrt{\sqrt{\sqrt{\sqrt{a}}}}$ and so on.
0
votes
1answer
65 views

Is there any simple analytic method for solving $\sqrt{x}+y=7$ and $x+\sqrt{y}=11$ simultaneously. [duplicate]

I am thinking of a nice and simple analytic method to solve the following equations simultaneously: $$\sqrt x+y=7;\\x+\sqrt y=11.$$ To my suprise I can't. But, I solve the system numerically using ...
-1
votes
1answer
44 views

Logic Question - Minimal Calculations Required [on hold]

Not sure were to start. Have fun.
1
vote
1answer
26 views

Basic graphing - plot v = 10i +4

So I have the function $v=10i+4$ where $v$ is the horizontal axis and $i$ is the vertical axis. Please excuse me for such a basic question but I can't work out how to draw this function. I figure if ...
5
votes
5answers
105 views

How $\sqrt{2}=1+\frac{1}{\sqrt{2}+1}$?

I have found it in the chapter about chain fractionals. I am unable to transform it to such state. $$\sqrt{2}=1+\sqrt{2}-1=?=1+\frac{1}{\sqrt{2}+1}$$
0
votes
3answers
69 views

Find x in this equation

Can you please help me find x in this equation? My knowledge on the rules of algebra is honestly limited. I simply cannot isolate the x on one side. $$\frac{(1+x)^3-1}{x}=3.1836$$
0
votes
1answer
25 views

Solution in terms of Lambert $W$ function

Is it possible to solve equation of the following form using Lambert $W$ function. $$(x-a)^2 = b(e^{-cx} - cx + 1).$$ If not, can it be solved using any other special function??
0
votes
1answer
16 views

Algebraic Equation Vexation

I was asked to help my sister with a bit of precalculus homework and completely drew a blank upon encountering this problem. I believe it was asking to "balance the equation, and set the answer to ...
3
votes
9answers
204 views

Why does $(a+b)^2= a^2+b^2 + 2ab$? Why is the $2ab$ there?

When I was doing research on finding the derivative I came across something strange. If $f(x) = x^2$ you find the derivative by going $$\frac{f(x+h)^2-f(x)^2}{h} =\frac{x^2+2xh+h^2-x^2}{h}.$$ Why ...
0
votes
1answer
34 views

What trig. identity would help solve $2 + \cos(2x) = 3\cos(x)$?

I need help with a homework question that has me puzzled. I need to solve the following equation: $$2 + \cos(2x) = 3\cos(x)$$ I don't see a good trig identity to apply. I tried $\cos(2x) = ...
4
votes
3answers
88 views

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

The Sum of ${11^{th}}$ power of the roots of the equation ${x^5+5x+1=0}$ ${My\; Try::}$ Let ${x=\alpha\;,\beta\;,\gamma\;,\delta\;,\mu}$ be the roots of the equation ${x^5+5x+1=0}$ So ...
0
votes
0answers
32 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
27 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: ...
1
vote
1answer
41 views

What are the steps to calculate the number of elements in a quotient?

Let $X = \{0,1,2,3,4,5,6,7,8,9\}$ and $ Y = \{0,2,4,6,8,9\}$. In $P(X) =$ power set of $X $ define the following relation: $$A R B \Leftrightarrow A \setminus Y = B \setminus Y $$ Then, how many ...
0
votes
2answers
40 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 ...
4
votes
2answers
23 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
79 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
45 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
48 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
33 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
45 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
44 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
32 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
26 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? ...
7
votes
1answer
72 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
33 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
38 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
78 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 ...
0
votes
0answers
16 views

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 ...
-1
votes
1answer
46 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
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 ...