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
votes
2answers
72 views

How to solve $\sin^3 x=\sin x\,$?

$\sin^3 x=\sin x$ I have absolutely no idea what to do. I've tried graphing, and I have a little better of an understanding, but I am at a loss.
0
votes
1answer
158 views

SAT arithmetic and pre-calculus

1.for the annual school fundraiser, santiago has p pledges each for c cents per lap that he jogs. If his school track has 4 laps per mile and santiago raises a total of d dollars, how many miles did ...
0
votes
4answers
158 views

Solving trigonometric equations to the fourth power.

$$\sin^4(x)-\sin^2(x)=0$$ My work: Let $t=\sin^2(x)$ Rewrite the original equation as: $t^2-t$ Factor: $t(t-1)$ $t=1$, $t=0$ What do I do from here?
0
votes
3answers
104 views

Simplifying and understanding inequalities in two variables

given an equation: $$\frac{x-y}{x+y}\ge0$$ What are the steps to simplify this into an understandable group of inequalities which will yield a solution set as a group of areas? I already know that $x+...
0
votes
5answers
72 views

Solving a quadratic trigonometric equation?

The equation is $6 \cos^2x+\cos x=1$, My work: $6x^2+x-1=0$ $(3x-1)(2x+1)$ $3x-1=0 ∨ 2x+1=0$ $x=\frac{1}{3} ∨ x= \frac{-1}{2}$ But I do not know how to progress further.
0
votes
2answers
102 views

Mathematical Induction Problem with Fraction

$$(3n-2)^2=\frac{n(6n^2-3n-1)}{2}$$ I can't seem to solve it out to the point where I can prove it right or wrong. I always hit some sort of roadblock where I don't have enough info to prove it wrong,...
2
votes
2answers
591 views

Solving for the principal value of a trigonometric equation

The question is as follows: $$3{\tan}^2(2x)-1=0.$$ Solve for $x$. What steps should I take to solve? The squaring is really throwing me off.
0
votes
2answers
80 views

Simplify: 2- {a(1-(1/2a)) /0.25}

Simplify: 2- {a(1-(1/2a)) /0.25} I removed the parenthesis by filling it in. I got: a(-a/2a). I multiplied a with 2a= 2a². So now I have 2a²/2a= a. Filling in a in the equation that I originally had ...
2
votes
1answer
86 views

Solving $|z-3| \leq|z-1-i|$

I was trying to solve graphicly: $$|z-3| \leq |z-1-i|$$ I plugged x and y in proper places as real componenets of the comlex number yielding in the end $-4x+2y+7 \leq0$ this might be tackled if ...
0
votes
1answer
205 views

How to derive this formula about the bracket function?

Is there a direct way of proving that $$ [nx] = [x] + [x+\frac{1}{2}] + [x+\frac{1}{3}] + \ldots + [x+ \frac{1}{n}]$$ for each real number $x$ and for each positive integer $n$? My effort: Let $...
2
votes
3answers
66 views

$\frac{2t-t^2}{t+2} \cdot (\frac{5t}{t-2} - \frac{2t}{t-2} )$

Simplify: $$\frac{2t-t^2}{t+2} \cdot \left(\frac{5t}{t-2} - \frac{2t}{t-2} \right)$$ I first subtracted the parenthesis because the denominator is equal. I then got: $$\frac{2t-t^2}{t+2} \cdot \...
1
vote
4answers
54 views

Express $3.72444\ldots$ as a fraction using the formula for geometric progressions

I did the following: $$3.72\overline{4} = 3.724+\left(\left(\frac{4}{10^4}\right)+\left(\frac{4}{10^5}\right)+\left(\frac{4}{10^6}\right)+\cdots\right)$$ where $a=3.724$ and $r=\dfrac{1}{10}$ Using $...
1
vote
2answers
40 views

If $a<b$, then $a^2<\frac{1}{3}(a^2+ab+b^2)<b^2$

I am trying to prove $a^2<\frac{1}{3}(a^2+ab+b^2)<b^2$ if $a<b$. I am having trouble with the left inequality: $a^2<\frac{1}{3}(a^2+ab+b^2) \Rightarrow 2a^2<ab+b^2$. If $a>0$, then ...
3
votes
3answers
150 views

System of quadratic equations with three variables

The problem is as follows: For $x,y,z \in R$, $$ \left\{ \begin{array}{l} x^{2} -yz-8x+7=0 \\ y^{2}+z^{2}+yz-6x+6=0 \end{array} \right. $$ What is the domain of $x$? One way to solve this is to ...
-1
votes
4answers
74 views

How can I factor $x^2 + 2\sqrt{3}\,x + 3$? [closed]

$$x^2 + 2\sqrt{3}\,x + 3$$ Anyone could tell me how may I factor this? Thanks a lot
1
vote
2answers
37 views

What will be the range of $f(x)= \frac{12}{\sqrt{(15-2x-x^2)}}$

Here's my try: Since the denominator involves a square root so I solved the following inequality: $15-2x-x^2>0$ which gives a solution set of $x=(-5,3)$. This is the domain of $f(x)$. However since ...
0
votes
1answer
69 views

My proof is wrong, can anyone tell me why?

$$\forall x \in \mathbb{Z}, \forall y \in \mathbb{Z}, [x(x+1) = y(y+1)] \Leftrightarrow [x = y]$$ $$\forall x \in \mathbb{Z} , \forall y \in \mathbb{Z}, [x(x+1)=y(y+1)]\Leftrightarrow [x=y]$$ $$\...
1
vote
1answer
17 views

system of binomial factors

A system of any number of binomial factors each the sum of two squares being supposed multiplied together , show , by any method, that their product is the sum of two squares.
0
votes
4answers
357 views

algebraic representation of a line in 3d

Is an algebraic representation of a line in 3d possible, or there can be only a parametric one?
0
votes
3answers
95 views

Map (-1500, 1500) on scale of (0, 100).

I've a minimum value = -1500 & maximum value 1500. Now, I've a scale of 0 to 100. How can I map my min & max value to this scale. That is, if I select 99 on scale, it should return 99% of ...
0
votes
1answer
33 views

the volume of pyramid value

when calculating the volume of pyramid using a determinnat, is it ok to take the determinanat in absloute value so that every negative result would be converted to positive volume number?
1
vote
3answers
175 views

If $\cos 25^\circ + \sin 25^\circ = k,$ then what is $\cos 20^\circ$?

Question: If $$\cos 25^\circ + \sin 25^\circ = k,$$ then what is $\cos 20^\circ$? What I did: I tried to square both sides, and obtained that $\sin 50 = k^2 -1$, however, this didn't get me ...
1
vote
2answers
53 views

$(18B^2/(A^2-9B^2)) - (A/A+3B) + 2$

Simplify: $$ \frac{18B²}{A²-9B²} - \frac{A}{A+3B} + 2$$ If the notation doesn't work like I wrote it above it's; Simplify: 18B^2/A^2-9B^2 - A/A+3B + 2. I made denominator common by expanding A²-9B²...
5
votes
2answers
114 views

$2^{1/4} \times 4^{1/8} \times 8^{1/16} \times 16^{1/32} \times \ldots\to2$

$2^{1/4} \times 4^{1/8} \times 8^{1/16} \times 16^{1/32} \times \ldots\to2$ How can I explain this to a school student who doesn't know what a limit is?
-1
votes
1answer
59 views

i am having problems with this operation.

Hello guys im trying to work out the follow operation with no success: $\left[\frac{x+5}{\left(x-9\right)\left(x+9\right)}+\frac{x+7}{\left(x-9\right)^2}\right]\cdot \left(\frac{x-9}{x-3}\right)^2+\...
1
vote
3answers
628 views

The high power integral

Im trying to solve the indefinite integral $$\int\frac{x}{(x^2+4)^3} \, \mathrm{d}x $$ I tried applying polynimial division and breaking to partial fractions but it didnt help...are there any other ...
0
votes
4answers
865 views

Using sum/product of quadratic roots to solve cubic equation

Given $\alpha$ and $\beta$ are the roots of the quadratic equation $6x^2 + 2x - 3 = 0$, how do I find the value of: $$ \alpha^3 + \beta^3 $$ and: $$ \frac{1}{\alpha^3} + \frac{1}{\beta^3} $$ ...
1
vote
3answers
38 views

Fraction exponents in division

if I have $\frac{a^{6/5}}{b^{1/5}}$, I know you subtract exponents when dividing so $6/5 - 1/5$ is $5/5$, so since that's just one, is this equal to $a/b$?
1
vote
1answer
39 views

How to solve this modulus question

IS there any solutions for this equation? ।x+3। - ।4-x। = ।8+x। Do we have to consider separately for each case like, ।x+3। <0 and ।x+3।>0 and go on?
0
votes
2answers
601 views

Linear Programming Problem?

You are about to take a test that contains computation problems worth 6 points each and word problems worth 10 points each. You can do a computation problem in 2 minutes and a word problem in 4 ...
5
votes
3answers
2k views

Finding a formula for specific limits

Hey guys I have a math problem I'm not sure how to go about solving. If someone could give me a systematic method for solving these kinds of problems that would be great. The problem is as follows: ...
3
votes
2answers
2k views

Limits of $\frac{\sin^2x}{x^2}$ as $x$ approaches infinity

I just want to make sure I'm on the right path with the problem. The problem is as follows: $$\lim_{x\to\infty}\frac{\sin^2x}{x^2}$$ I rewrote it as follows: $$\lim_{x\to\infty}\frac{(\sin x)^2}{x^...
2
votes
1answer
375 views

Solving Weird Exponential Equations

I am working on my math homework when I encountered a difficult problem. I simplified the equation and substituted smaller numbers to get this: $n*2^n>10$ I have tried standard algebraic ...
0
votes
2answers
155 views

Labeling derivatives of functions from a graph

I have a question about derivatives and identifying them on a graph. I came across a problem that looks like this: The figure shows graphs of f, f-prime, f-prime-prime, and f-pime-prime-prime. ...
1
vote
1answer
107 views

Solving a second level functional equation over all functions $g$

I am trying to find a closed form expression $f$ such that $$f(g(x+1) - g(x)) + f(g(x) - g(x-1)) = f(g(x))$$ For all functions $g$ I have concluded that for polynomials $$2^{n+1}f(0) = f(a_0 + ...
1
vote
2answers
149 views

How to show $\log_2 x \cdot \log_{0.25} x \cdot \log_{0.125} x \cdot \log_{16} x > \frac {2}3$?

I was trying to solve $\log_2 x \cdot \log_{0.25} x \cdot \log_{0.125} x \cdot \log_{16} x > \frac {2}3$ and I keep getting a partial answer of $x>4$ though answer key suggests a more expanded ...
0
votes
1answer
79 views

Simplify $\frac{1}{x-2}-\frac{1}{x+2}$

Simplify: $$\frac{1}{x-2}-\frac{1}{x+2}$$ What I did was multiply both sides to get the denominator equal: $$\frac{x+2}{(x-2)(x+2)}-\frac{x-2}{(x-2)(x+2)}=\frac{x^2+4x+4}{x^2-4} =\frac{4x+4}{ -4}=\...
1
vote
2answers
1k views

solving equations with powers

Im trying to solve the equation $$3\cdot2^{-2/x} + 2\cdot9 ^{-1/x} = 5\cdot6^{-1/x }$$ So far I tried applying logaritmas but it didnt prove helpful...are there any other ways?
0
votes
6answers
77 views

solving the inequalty

are there any ways to solve :$ x^4 -6x^3 +28x^2 -64x +96 >0$ ?
2
votes
4answers
162 views

How to solve: $\cos^2x + \sin x = 1$

$\cos^2x + \sin x = 1$ How to solve for $x$?
0
votes
2answers
252 views

Quarters weigh 6 grams while dimes weigh 2 grams.

Quarters weigh $6$ grams while dimes weigh $2$ grams. Tiffany has $\$5.35$ worth of quarters and dimes in her pocket weighing a total of $124$ grams. How many quarters does Tiffany have?
-1
votes
3answers
140 views

Can there be more than one power series expansion for a function.

I guess the answer is NO, for polynomials. I know that there are more than one series expansion for every function. But I am talking about power series here. All Ideas are appreciated
0
votes
3answers
90 views

How to prove $e^x=\lim_{n\to\infty}\left(1+\dfrac{x}{n}\right)^n$

Define $$e=\lim_{n\to\infty}\left(1+\dfrac{1}{n}\right)^n,$$ how to prove that $$e^x=\lim_{n\to\infty}\left(1+\dfrac{x}{n}\right)^n,$$ for all $x$. Thank you.
0
votes
1answer
76 views

A cyclist rides from Alberville to Bakersville

A cyclist rides from Albertville to Bakersville, a distance of 120 km. His return trip takes 1 hour longer, because his speed decreases by 10 km/hr. How fast does he ride each way?
0
votes
1answer
71 views

A square pool is surrounded by a concrete deck

A square pool of area $144 \, \text{m}^2$ is surrounded by a concrete deck of area $25 \, \text{m}^2$. What is the perimeter of the outside of the deck?
1
vote
4answers
150 views

Simplification of Terms $ 2^{(1/\ln2)}$

The final answer is $\mathrm e$. Can anyone show steps? $ 2^{(1/\ln2)}$ I tried log laws without any result. I'm pretty sure I'm stuck somewhere really easy.
0
votes
2answers
53 views

Radius of curvature and continuous functions

Let $\kappa (x)$ be radius of curvature function for a continuous function $f(x)$. Is it necessary that $\kappa(x)$ will have extrema when $f(x)$ does. And the nature of extrema is opposite to that ...
0
votes
2answers
27 views

is my linear equation's result correct?

I have the following linear equation (/ is of fraction symbol): 5x + 3/4 = 6x + 6(2/4) And my result is: ...
2
votes
1answer
120 views

Finding intervals on the graph of a function

I'm having a little trouble with a problem that seems suspiciously easy and I wanted to make sure I'm on the right path. The problem reads as follows: The rate of r (in liters per minute) at which ...
1
vote
1answer
747 views

Equation of parabola, tangent at vertex [closed]

Two tangents on a parabola are $x-y=0$ and $x+y=0$. If $(2,3)$ is the focus of the parabola, then find the equation of tangent at the vertex. Thanks. My thoughts: Can't figure out anything :(