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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4answers
37 views

Find $\tan x $ if $\sin x+\cos x=\frac12$

It is given that $0 < x < 180^\circ$ and $\sin x+\cos x=\frac12$, Find $\tan x $. I tried all identities I know but I have no idea how to proceed. Any help would be appreciated.
1
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4answers
40 views

Solve for $x$, $\tan x +\sec x = 2\cos x$ ; $−∞ < x < ∞$

Solve for $x$, $\tan x +\sec x = 2\cos x$ ; $−∞ < x < ∞$ $$\tan x + \sec x = 2\cos x$$ I tried changing it all to sin and cos $$\frac{\sin x}{\cos x} + \frac{1}{\cos x} = 2\cos x$$ then I ...
0
votes
2answers
32 views

Express $\sin(x) + \sqrt{3}\cos(x)$ in the form $A\sin(x + a)$ [on hold]

How would I go about expressing $\sin(x) + \sqrt{3}\cos(x)$ in the form $A\sin(x + a)$, where $A > 0$ and $0 < a < \pi/2$?
2
votes
2answers
98 views

Work with algebra

$2{x}^2 + 3{y}^2 = {a}^2$ What is the maximum value of $3x+2y$ ? Not knowing calculus and how to graph a ellipse may make this harder. But is there a way to get around calculus? Please determine the ...
3
votes
3answers
72 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 ...
0
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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 ...
0
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4answers
34 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
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0answers
26 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
88 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 $3x+2y$?
10
votes
2answers
408 views

Intuitive ways to get formula of cubic sum

Is there an intuitive way to get cubic sum? From this post: combination of quadratic and cubic series and Wikipedia: Faulhaber formula, I get $$1^3 + 2^3 + \dots + n^3 = \frac{n^2(n+1)^2}{4}$$ I think ...
1
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0answers
43 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 $b=x^x$?
3
votes
1answer
37 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
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} ...
0
votes
1answer
27 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
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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 ...
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 ...
0
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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.
2
votes
2answers
87 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
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!
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 ...
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? ...
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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
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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 ...
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
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 ...
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.
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 ...
1
vote
3answers
42 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. ...
1
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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. ...
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 ...
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 ...
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: $$\begin{cases} a+b=0 \\ a-b=1 \end{cases}$$
0
votes
4answers
68 views

Questions about solving inequality: $2 < \frac{3x+1}{2x+4}$

Solve the inequality: $2 < \frac{3x+1}{2x+4}$ Step 1: I simplified $\frac{3x+1}{2x+4}$ into: $3x+1-2x-4= x-3$. Step 2: $2>x-3$ Here I subtracted $2$ from both sides into: $x>-5$ or ...
0
votes
0answers
17 views

Can I evaluate polynomials with prime numbers to find possible irreductible factors?

Let $p(x,y)$, $c(x,y)$ and $d(x,y)$ be two variable polynomials with integer coefficients which satisfy $p(x,y)=c(x,y)\cdot d(x,y)$. Given $m, n$ positive prime numbers and given $e(x,y)$ another ...
1
vote
2answers
51 views

Inequality - Find what value of $t$ satisfies: $ (t/24) - (t+1) + (3t/8) < (5/12) (t+1)$

Inequality - Find what value of $t$ satisfies: $(t/24) - (t+1) + (3t/8) < (5/12) (t+1)$. Step 1: I multiplied both sides by $24$ and divided to get: $t-24(t+1)+9t < 10+24(t+1)$. Step 2: I ...
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
3answers
109 views

Simplify tan$\theta$ cos$\theta$

How do I simplify tan$\theta$ cos$\theta$ ? Why is this so hard to do? What pieces of information should I know before doing these? Can someone just tell me were am I going wrong? I have 5 days ...
1
vote
2answers
86 views

limit of a rational function, $\lim _{x\to 1} \frac{x^3 +x-2}{1-x}$ [on hold]

$$\lim _{x\to 1} \frac{x^3 +x-2}{1-x}$$ can you please tell how to solve this ?! I tried too many things, non worked ! it is not zero for sure ! mathway gave me $-4$ (no steps)
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): ...
0
votes
3answers
50 views

Simplify $\tan(360 - \theta)$

I am aware that $\tan(\alpha-\beta)=\dfrac{\tan(\alpha)-\tan(\beta)}{1+\tan(\alpha)\tan(\beta)}$ So for my question: $\tan(360 - \theta)$ Do I choose random value for $\theta$ and plug it into the ...
1
vote
2answers
35 views

Number as sum of multiples of 3 and 5.

There were 10 questions on a test. A student gets 5 points for every correct answer and 3 points for every partially correct answer. If the student got 19 points, how many correct and partial ...
0
votes
3answers
56 views

Simplify $\sin (90 - \theta)$

Title. I have no idea what to do. Is their an identity I have to remember? What am I supposed to do to the equation? Do I have to solve for something first, what does it mean by simplify?
-2
votes
2answers
26 views

Solving Linear equation Gaussian method of elimination [on hold]

Solving linear equation Gaussian method of elimination $$\left\{ \eqalign{ -x-y & =7 \\ 4x-y & =-3 } ...
0
votes
1answer
28 views

Reverse a formula with codependent expressions

I apologize for my title, but I really am a long way from understanding how to even describe my problem accurately, let alone solving it. I'm looking to reverse this formula: ...
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, ...