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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3
votes
2answers
57 views

Find the value of this infinitely nested radical (it appears to obtain multiple values)

Find the value of $$\sqrt{1-\sqrt{\frac{17}{16}-\sqrt{1-\sqrt{\frac{17}{16}-\cdots}}}}$$ This is not as simple as it looks for one reason - there are $2$ real solutions to the equation ...
1
vote
4answers
53 views

Trigonometric functions of the acute angle

Find the other five trigonometric functions of the acute angle A, given that: \begin{align} &\text{a)}\ \ \sec A = 2 \\[15pt] &\text{b)}\ \ \cos A = \frac{m^2 - n^2}{m^2 + n^2} \end{align} ...
1
vote
3answers
40 views

Finding Horizontal/Oblique Asymptote of $y=\frac{\sqrt{x}+1}{\sqrt{x}-1}$

Is it correct to simply subsitute $\sqrt{x}$ with $x$ when finding horizontal or oblique asymptotes? The method works but I am not sure if it is formally sound enough to pass muster in an examination. ...
0
votes
3answers
42 views

How to memorize the trigonometric identities?

I am stuck trying to memorize the trig identities, and try as I may, I just can't get them to stick (especially the sum-product and product-sum formulas). I am concerned I won't be able to memorize ...
1
vote
1answer
28 views

Evaluation of a Hankel-like determinant

I consider the following determinant (Hankel-like?) $$ [f_1,f_2,...,f_n]:=\begin{vmatrix} f_1 & f_2 & \cdots & f_{n-1} & f_n\\ n-1 & f_1 & \cdots & f_{n-2}& f_{n-1}\\ 0 ...
0
votes
4answers
33 views

Given a satisfactory real number = [any integer]/(2b) where a and b are integers, how would one find the minimum value of b?

For instance, 0.625 = 5/(2*4). Given 0.625, how would one find 4? 0.75 = 1/(2*2). Given 0.75, how would one find 2? I should ...
0
votes
0answers
25 views

How to convert formulas: Convert cost/revenue as a percentage to revenue/cost as a percentage

Is it possible to convert cost per acquisition percentage (cost/revenue x 100) to return on spend percentage (revenue/cost x 100) where cost and revenue are variable? I use cost per acquisition (CPA) ...
3
votes
1answer
86 views

A unfamiliar question

I'm sure asking this kinda problem is stupid but somehow I have never seen such problems before. $2{x}^2 + 3{y}^2 =0$ what is $3x+2y$?
1
vote
3answers
42 views

“Full”-Simplification of arbitrary mathematical expressions

I've come across many (classroom) problems, like Roy did, whereby the solution to a problem, $$−3(7−2x)^2−5(1+x)^2$$ is the result of simplifying that expression as much as is possible, i.e. ...
3
votes
1answer
36 views

Simplify $\frac {\sqrt5}{\sqrt3+1} - \sqrt\frac{30}{8} + \frac {\sqrt {45}}{2}$

I am trying to find the value of: $$\frac {\sqrt5}{\sqrt3+1} - \sqrt\frac{30}{8} + \frac {\sqrt {45}}{2}$$ I have the key with the answer $\sqrt 5$ but am wondering how I can easily get to that ...
1
vote
0answers
42 views

How prove: $a=x$ and $b=x^x$ for $x^{a+b}=a^b b$?

Let $x, a, b$ natural numbers such that $x^{a+b}=a^b b$. How prove: $a=x$ and $b=x^x$?
1
vote
1answer
23 views

Simple question (hopefully) on unitary method

In India we have an exam called NEST. I gave it today, and this was a question I encountered: Lactobacillus sp. and Streptococcus sp. are two bacterial species responsible for curdling milk. One ...
0
votes
1answer
26 views

Acute angle and trigonometric functions

Given that $\theta$ is an acute angle and $\cos\theta = \dfrac{7}{25}$. Find: $\tan\theta$, $\sin\theta$, $\sec\theta$.
1
vote
4answers
32 views

Number of distinct real roots with $e^{-x}$ in the equation

How to find the number of distinct real roots of the equation $$13x^{13}-e^{-x}-1=0$$ I know that we generally find number of real roots by observing number of sign changes in $f(x)$ and $f(-x)$ but ...
32
votes
19answers
16k views

Is $.999999999… = 1$?

I'm told by smart people that $0.999999999... = 1$ and I believe them, but is there a proof that explains why this is?
0
votes
2answers
15 views

Do you use the inner or outer degrees of a right-angled triangle when calculating the vertical component of a vector

I'm given the assignment of finding the vertical component of vector a - b. Below is an image of vector A. ||A|| = 6. I need to calculate side y. I was following the assignment and tried solving y ...
3
votes
4answers
287 views

Could somebody please help me prove this using the properties of real numbers introduced in elementary algebra?

I would like to prove the following useful property of real numbers: For every real number $a$, prove $\frac{a}{1} =a,$ where for all real numbers a and b: $\frac{a}{b}=a*\frac{1}{b}; b\neq0,$ and ...
1
vote
4answers
1k views

What's wrong with my aproach to solving this equation with multiple logarithms?

A question I was faced with asked "For which $x$ is $\log_{10}(x)^{\log_{10}(\log_{10}(x))}= 10,000$?" My instincts tell me I can say $$\log_{10}(x)=10$$ and $$\log_{10}(\log_{10}(x))=4$$ However, ...
2
votes
2answers
85 views

Evaluation of $\int\frac{\sqrt{\cos 2x}}{\sin x}dx$

Evaluation of $$\displaystyle \int\frac{\sqrt{\cos 2x}}{\sin x}dx$$ $\bf{My\; Try::}$ Let $\displaystyle I = \int\frac{\sqrt{\cos 2x}}{\sin x}dx = \int\frac{\cos 2x}{\sin^2 x\sqrt{\cos 2x}}\sin xdx ...
0
votes
3answers
60 views

How to solve for $x$ for $\frac{1}2 x^{-1/2}- \frac14x^{-3/4}$

This is a derivative and I am trying to find the max and min. Right now I am trying to solve for x. $$\frac{1}2 x^{-1/2}- \frac14x^{-3/4}$$ $$\frac{1}{2 x^{1/2}}- \frac1{4x^{3/4}}$$ $$\frac{1}{2 ...
0
votes
0answers
20 views

why does a polynomial with square values work like this?

If $q(x)$ is a polynomial and $q(n)$ is a square for all integers $n\geq n_0$, then $q(x)$ is square of a polynomial.
1
vote
1answer
1k views

How to find the width of a path around a rectangle, given the area of the path?

A man built a walk of uniform width around a rectangular pool. If the area of the walk is 165 square feet and the dimensions of the pool are 17 feet by 11 feet, how wide is the walk? How should I ...
0
votes
1answer
35 views

In which direction to round the answer, if it represents maximal population that could be infected?

I'm calculating the maximum number of a population that could be infected. I have an answer of $240.0729395$. Should I round this to $241$ because that is the max? Everyone else says $240$ is ...
0
votes
1answer
914 views

What is the greatest speed he can reach with an acceleration of 5.00 g before blacking out?

A jet fighter pilot wishes to accelerate from rest at $5.00$ G to reach Mach-3 (three times the speed of sound) as quickly as possible. Experimental tests reveal that he will black out if this ...
3
votes
2answers
67 views

An inequality I am stuck on

This is somehow related to this problem but I don't have any idea about it. $a,b,c,d$ are positive reals such that $a+b+c+d=4$ $$\frac{1}{a+3}+\frac{1}{b+3}+\frac{1}{c+3}+\frac{1}{d+3}\le ...
0
votes
1answer
20 views

Using $\pm$ to express “in the range of” statement.

Does it make sense to write: $x = \left\{A\pm B\right\}$ To mean that $x$ falls in the range of $\left\{A-B,A+B\right\}$? If not, what would be the correct way of expressing this? Many thanks!
2
votes
5answers
3k views

Simple Proof by induction: “9 divides $n^3 + (n+1)^3 + (n+2)^3$”

I'm trying to prove using induction that 9 divides $n^3 + (n+1)^3 + (n+2)^3$ whenever $n$ is a non-negative integer. So far, I have: Base case: P(1) = (1) + (8) + (27) = 36, 36 can be divided by 9 ...
1
vote
3answers
30 views

Factoring when differentiating expressions

I'm having trouble with differentiating a expression. I do it one way, wolfram alpha does it another. Let me show you what I mean. The original expression is this: $$\frac{1}{2u^3}$$ I start by ...
2
votes
4answers
176 views

Solve The Triangle

I am having a tough time trying to solve this problem. I have utilized the 30, 60, 90 triangle measures for the length of sides. However, I am stuck since the side that would be √3 has 100 as its ...
4
votes
1answer
16 views

Solving Differential equations with Laplace transform

$\displaystyle y''+4y'+3y=e^{-t}$, given $\displaystyle y(0)=y'(0)=1$ My Attempt: Taking Laplace transforms on both sides $\displaystyle $ $\displaystyle [s^2\bar y-sy(0)-y'(0)]+4[s\bar ...
0
votes
1answer
24 views

Questions Number 8 and 9 from College Algebra CLEP Study Guide

Needing help with some more questions on my College Algebra Clep Study Guide. :/ I'm not looking for answers. I'm just looking to know how to solve some questions. Can anyone please help? ...
1
vote
1answer
40 views

Quadratic inequality with parameter

Hi I've got this inequality with parameter $a\in R$ $\frac{x+a}{x}\le x+2$ I've solved it but I've got different results than book. I've done it by dividing it into 2 cases. 1. x<0 2. x>0 and then ...
0
votes
3answers
40 views

Trigonometric Identities help

How do you solve this? I can't figure out what I should do. $$\sin ^4\left(A\right)+\cos ^2\left(A\right)=\cos ^4\left(A\right)+\sin ^2\left(A\right)$$ Also, why is this equal zero? Can someone ...
0
votes
2answers
20 views

Question Number 5 from College Algebra CLEP Study Guide

Needing help with some questions on my College Algebra Clep Study Guide. :/ I'm not looking for answers. I'm just looking to know how to solve some questions. Can anyone please help? ...
1
vote
3answers
24 views

Solving quadratic equations without the quadratic forumla

Is it possible to solve the following equation, in terms of $q$, without using the quadratic formula? $t - (m-q)^2 = v - (m-p)^2$ I asked a similar question this morning (about the quadratic ...
2
votes
4answers
59 views

Determined or not?

the function $\dfrac {2x}{3x-\sqrt{x} }$ is not derterined for values of $x$ equale or samller than zero, though when I take the limit $ \lim_{x \to 0^+} \dfrac {2x}{3x-\sqrt{x} }$ the output is zero ...
0
votes
0answers
26 views

Laplace transform - Heaviside algebra

I'm strugling with some algebra around a laplace transform with heaviside. The start function is $L(2tH(1-t)) + L(2H(t-1))$ so from this, I'm supposed to convert it to $L(2t) + L(2(1-t)H(t-1))$ ...
1
vote
3answers
31 views

How to find the general form (Ax-By-C = 0) of a line with an undefined slope

This is how the question reads: "The equation of the line that goes through the points (3, -6) and (3, 10) can be written in general form Ax + By + C = 0 where A = _ B = _ and C = ____" I know the ...
2
votes
3answers
33 views

How do I find the domain of this function

I would like to know which operations i have to do to get the domain of this function: $$y=\sqrt{\frac{1}{x}-1}$$ I have researched and the solution of the inequality $\frac{1}{x}-1 \geq 0$ is ...
9
votes
6answers
600 views
+50

How find the value of the $x+y$

Question: let $x,y\in \Bbb R $, and such $$\begin{cases} 3x^3+4y^3=7\\ 4x^4+3y^4=16 \end{cases}$$ Find the $x+y$ This problem is from china some BBS My idea: since ...
1
vote
1answer
25 views

rearrange $t - (m-q)^2 = v - (m-p)^2$ for quadratic formula form $ax^2 + bx +c = 0$ solving for $q$

I have the equation $t - (m-q)^2 = v - (m-p)^2$ which I would like to rearrange to be able to apply the quadratic formula, and solve in terms of $q$. Accordingly, it needs to be in the form: $ax^2 ...
0
votes
0answers
16 views

Rescaling of $|x_n - n|<1+\epsilon$ [on hold]

Good morning, I would like I would like to know a rescaling of $$|x_n - n|<1+\epsilon$$ with $0<\epsilon<1$, $x_n\in \mathbb R$, $n=0,\pm 1, \pm 2, \pm3, ...$ in this way: $$|x_m - ...
1
vote
3answers
37 views

solve $-(x_m - x_q)^2 = -(x_m - x_p)^2$ in terms of $x_q$

I have an equation, $-(x_m - x_q)^2 = -(x_m - x_p)^2$ which I want to solve in terms of $x_q$. I can see (by using a number line) that $q$ can have two solutions: $x_q = x_p$ or: $x_q = 2x_m-x_p$ ...
-1
votes
0answers
19 views

Calculate distance between two objects based on their visible height for a specific focal length

How do I calculate the distance between to objects of the same size base on their height for a given focal length. Both object 1 and object 2 are 15 cm in height (actual size). Object 2 looks ...
5
votes
13answers
9k views

how to find center of an arc given start point, end point, radius, and arc direction?

Given an arbitrary arc, where you know the following values: start point (x0,y0), end point (x1,y1), radius (r) and arc direction (e.g. clockwise or counterclockwise from start to end), how can I ...
0
votes
4answers
57 views

Simplify $ (a^{-2} - b^{-2})/(a^{-1}-b^{-1})$ [on hold]

Simplify $$ (a^{-2} - b^{-2})/(a^{-1}-b^{-1}).$$ My answer is $\frac{1}{a+b}$. Am I correct? I had a precalculus final today and remembered this is a question.
1
vote
1answer
50 views

Finding cut-off point for utility function

OK, so apologies for the easy question, but I'm new to this! This is somewhere between elementary algebra, and beginner's game theory. The question comes from a paper I read here (see p. 193): ...
1
vote
1answer
40 views

Find value of $x$ for: $(1/3)(1-x) \geq 2(x-3)$

Find what value of $x$ satisfy: $(1/3)(1-x) \geq 2(x-3)$ First I multiplied both sides by $3$ so that $1/3$ became $3/3=1$. So I tried to find $x$ this way: $(1-x) \geq 6(x-3)$. I tried solving it ...
0
votes
2answers
18 views

Combining Functions Question

Question: If $f(x)=x^2-x+2$ and $g(x)=x-2$, find $h(x)$ such that $f(x)=g(h(x))$ I am not sure if I am on the right track here so far, I came to this mostly through guess and check, perhaps there is ...
0
votes
3answers
38 views

Factoring positive rational numbers

I don't understand how he goes from this: $$2n(a+b)+a-b=1\,\,\,; \forall n\in \mathbb{N^*}, \,\,\, \mathbb{N^*}=\{1,2,3,...\}$$, To this: $$\begin{cases} a+b=0 \\ a-b=1 \end{cases}$$