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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3answers
171 views

Temperature as a quadratic function; when does it reach zero?

First of all, this is a very silly question, but I finished high school a long time ago and I really don't remember much about some basic stuff. I have: $$T(x) = -\frac1{10}x^2 + \frac{24}{10}x - ...
1
vote
1answer
1k views

How to Convert a Number to Roman Numerals

i don't How to Convert a Number to Roman Numerals Using mathematical equation. if M=1000,D=500,C=100,L=50,X=10,V=5,I=1 then how convert any decimal number to Roman Numerals? such as if 1952 then its ...
1
vote
2answers
122 views

Why does $\frac{f(x)}{\frac{f(y)}{h}} = \frac{f(x)}{hf(y)}$

Earlier, I asked a similar question, but this one's getting me confused, as well. Why does $\frac{a}{\frac{b}{x}} = x \times \frac{a}{b}$, but the following expression holds true? $$ ...
2
votes
8answers
238 views

Why does $\frac{a}{\frac{b}{x}} = x \times \frac{a}{b}$?

As much as it embarasses me to say it, but I always had a hard time understanding the following equality: $$ \frac{a}{\frac{b}{x}} = x \times \frac{a}{b} $$ I always thought that the left-hand side ...
0
votes
1answer
79 views

Please help with this word problem involving trigonometry [closed]

A small aircraft has a glide ration of 15:1. (The glide ratio means the plane moves 15 units horizontally for every one unit elevation). You are exactly in the middle of a 3.0 mile diameter lake at ...
2
votes
1answer
115 views

Solve the equation using logarithms

Equation $$ e^{2x+1.21} = 114\cdot 4^x $$ steps I've done so far. $2x + 1.21 = \ln(114) \cdot \ln(4) \cdot x$ $x = (\ln(114) * \ln(4))/1.21$ I don't think I was allowed to move the $x$ from the ...
2
votes
3answers
90 views

Solve $33(1.207)^x = 47(1.547)^x$

How can we solve this equation? $$33(1.207)^x = 47(1.547)^x$$ I've done this step so far.$$33x\ln(1.207) = 47x\ln(1.547)$$ This is where I get stuck. Won't moving an $x$ to one side eliminate all ...
2
votes
1answer
67 views

Solve the equation by using logarithms

The equation is $64(12)^{8x} = 195$ the steps I have done so far. $12^{8x} = 195/64$ $8x \cdot \ln(12) = \ln(195/64)$ $8x = (\ln(195/64))/(\ln(12))$ not sure how to divide $8x$ to get $x$ by ...
1
vote
0answers
61 views

Interest % and final payment

I dont get these two questions? How should I calculate the answer? A) Sam intends to retire in 6 yrs. To supplement his pension he would like to receive $\$234$ every 2 weeks for 10 yrs. If he is ...
1
vote
1answer
86 views

An algebra problem related to trigonometry

Let $\alpha $ be an acute angle such that $\sin \alpha = \dfrac{3 \sqrt 3}{14}$. Prove that $$\frac{2\cdot7^{n}}{\sqrt 3}\sin \left(n\alpha + \dfrac{\pi}{3} \right) \in \mathbb{Z} \qquad\forall ...
3
votes
3answers
188 views

Finding the minimum of $\frac pq + \frac rs$ for distinct integers $p, q, r, s$ from $\{1,2,3,4,5,\ldots,16,17\}$

Here is the question: Four distinct integers $p$, $q$, $r$ and $s$ are chosen from the set $\{1, 2, 3, 4, 5, \ldots, 16, 17\}$. The minimum possible value of $\frac pq + \frac rs$ can be written ...
2
votes
0answers
206 views

how to obtain state space diagram and state space model for transfer function

How do we obtain the state space diagram and state space model for transfer function for example the question is given How to draw state variable diagram for the given transfer function ...
2
votes
3answers
143 views

A five-digit number N…

A five-digit number $N$ is equal to $45$ times the product of its $5$ digits. Find $N$. Please help. I am not sure how to solve this. I have a feeling it is simple
0
votes
1answer
25 views

Weekly Compound Interest on $\$480$, $7\% $ interest, for $12$ years

So far I have the equation $480\left(1+\cfrac{0.07}{52}\right)^{52\times12}$ I think that is the right equation but it is not giving me the correct answer.
4
votes
1answer
41 views

$(a - 1)x^2+3(a + 1)x+4(a - 1) = 0$ has real solutions iff $7a^2 - 50a + 7\leq 0 $

How can we show that $(a - 1)x^2+3(a + 1)x+4(a - 1) = 0$ has real solutions if and only if $7a^2 - 50a + 7\leq 0$? I know these are quadratics and can solve them, but I'm not entirely sure what the ...
3
votes
2answers
98 views

how do you express (a + b) to the exponent 0.25 in terms of a and b separately?

how do you express (a + b) to the exponent 0.25 in terms of a and b separately? For instance (2 + 4) to the 0.25 = 1.5651 but how would you solve (2 + unkown) to the 0.25 = 1.5651?
0
votes
1answer
41 views

Showing an inequality is valid

Suppose that $a$, $b$ and $c$ are positive real numbers and that $a \leq b + c$. By cross multiplying or otherwise, show that $$\frac{a}{1 + a} \leq \frac{b}{1 + b} + \frac{c}{1 + c}$$ Anyone able ...
6
votes
6answers
1k views

How to simplify $\frac{4 + 2\sqrt6}{\sqrt{5 + 2\sqrt{6}}}$?

I was tackling through an olympiad practice book when I saw one of these problems: If $x = 5 + 2\sqrt6$, evaluate $\Large{x \ - \ 1 \over\sqrt{x}}$? The answer written is $2\sqrt2$, but I ...
4
votes
5answers
205 views

Factor $1+x+x^2+x^3+…+x^{14}$

In a previous task, I was asked to factor $1+x+x^2+x^3$ for $x \in \mathbb{R}$, which I accomplished by solving $1+x+x^2+x^3 = 0 \to $ $1+x(1+x+x^2) = 0 \to $ $x(1+x+x^2) = -1$ which has a ...
1
vote
1answer
243 views

How to get original number when 4% has been subtracted

I need to reverse the effect of subtracting 4% from a large anmount of numbers. One example: There was subtracted 4% from 100 making it 96. What kind of formula would make it 100 again? UPDATE: ...
0
votes
0answers
51 views

Proof of Bijection [duplicate]

Possible Duplicate: Construct a bijection between $\mathbb{Z}^+\times \mathbb{Z}^+$ and $\mathbb{Z}^+$ Recall that $\Bbb N = \{0, 1, 2, \dots\}$ is the set of whole numbers. Let $f$ be a ...
1
vote
1answer
270 views

Equation of a circle with specific conditions

I am trying to find the equation for a circle that has the center at (-1 , 4) and passed through the point (3, -2) This seems like a straightforward problem. $$(x+1)^2+(y-4)^2 = 16$$ The radius ...
1
vote
5answers
483 views

Finding a real solution to a quadratic

I am suppose to find all the solutions to this problem, I think some theorem states that there can only be as many solutions to the problem as the highest degree. I know that calculus reinforces this ...
0
votes
3answers
104 views

How to multiply out $(\sqrt{2}) (\sqrt{2}i)(\sqrt{2}+\sqrt{2}i)$

How to multiply out $(\sqrt{2}) (\sqrt{2}i)(\sqrt{2}+\sqrt{2}i)$? $i =$ the complex imaginary number.
2
votes
2answers
290 views

Rationalizing the denominator of $\frac {\sqrt{10}}{\sqrt{5} -2}$

I have the expression $$\frac {\sqrt{10}}{\sqrt{5} -2}$$ I can't figure out what to do from here, I can't seem to pull any numbers out of either of the square roots so it appears that it must remain ...
1
vote
1answer
302 views

Simplifying third degree polynomials

I am trying to figure out this simple problem but I have no idea how to do it. $\frac {2x^3 - 5x^2 -4x -3}{2x^3 + x^2 - 18x -9}$ I know that I can keep guessing at random to find a factor in the ...
0
votes
3answers
76 views

How to calcualate how many unique set of 6 can i have in a given set.

Hello my question is quite simple i would think but i just cant seem to find an answer. I have a set of $\{1,2,3,4,5,6,7,8,9,10\}$ and i would like to calculate how many unique given sets of $6$ can i ...
12
votes
3answers
557 views

Showing $\sqrt{2}\sqrt{3} $ is greater or less than $ \sqrt{2} + \sqrt{3} $ algebraically

How can we establish algebraically if $\sqrt{2}\sqrt{3}$ is greater than or less than $\sqrt{2} + \sqrt{3}$? I know I can plug the values into any calculator and compare the digits, but that is not ...
2
votes
2answers
62 views

Percentages, Points, and a Square

I know this is probably an easy problem, but I can't for the life of me figure it out. My problem is slightly hard to explain with words alone, so a picture can be found here. Point $a = (0, 0)$, ...
6
votes
3answers
204 views

What operations on an equation cause it to be destroyed?

I approached my calculus professor about something he said which didn't make much sense to me - He says that in the process of calculating $\lim_{x\to\infty} f(x)^{g(x)}$, you can convert it to ...
0
votes
2answers
160 views

Simplify $\frac{5+ \sqrt{20}}{6+ \sqrt{20}}$ to $\frac{5+ \sqrt{5}}{8}$

I would like to know how the following expression was simplified? From this $$\frac{5+ \sqrt{20}}{6+ \sqrt{20}}$$ to this $$\frac{5+ \sqrt{5}}{8}$$
4
votes
2answers
1k views

Direct sums and direct products

This question has been in my head for a while. And today it appears again when I am reading Arveson's book on $C^*$-algebras. He says Countable direct products of Polish spaces are Polish. ...
1
vote
1answer
85 views

Solve Equation of three variables

I want to solve this system equations. I found 2 solutions, but it's was effort algebra with the second equation and take $z=0$ in (1), you can see 2 solutions with $x=-2,2$ and $ y=1,-1$. $$ (1) \ ...
2
votes
1answer
51 views

Proportionally scale values

I have a series of values that are percentages of a whole For example: 10 % 8 % 17 % 14 % 12 % 19 % 16 % These are plotted on a bar graph with the top scale being 100%. As you can see there is ...
1
vote
1answer
65 views

Find a function f(x) such that the parametric curve could be obtained by flipping the graph

Find a function f(x) such that the parametric curve could be obtained by flipping the graph of f across the line with slope 1 that goes through the origin. parametric curve with coordinates (t^14, ...
0
votes
2answers
40 views

Ratio for home work

In the year 2000 a rich aunty won £12000 and decided to share her money between her 3 nieces in the ratio of their age 14 10 18. she died in 2006 how much will each niece get?
0
votes
1answer
360 views

NMAT Practice Exam Math Problem: $3^{n+2}+(3^{n+3}-3^{n+1}) =~?$

The problem is from an NMAT Practice Exam. The problem is multiple choice. It looks easy enough... $$3^{n+2}+(3^{n+3}-3^{n+1}) =~?$$ a.) $\dfrac1{3^{n+1}}$ b.) $\dfrac1{3^{n+2}}$ c.) $\dfrac38$ ...
0
votes
1answer
237 views

Solve Leonardo's purse problem: three merchants find a purse in road

I've been reading the, 'The Man of Numbers' book by Keith Devlin about Leonardo of Pisa who wrote the historic book on arithmetic around 1202 CE. One of the problems, on page 70, goes as follows: ...
3
votes
4answers
73 views

How much detail should be shown when writing out a problem?

When a somebody asks you to solve a problem and "show your work," how much detail should be shown? For a simple example, if I were to solve $5x+7=17$, should I do: $$5x+7=17$$ $$5x+7-7=17-7$$ ...
0
votes
1answer
156 views

help me to Solve a trick equation

I am trying to solve a trick question that my teacher gave to me. This question is related to linear algebra. I have no idea on where to start with this question. I need some guidance in getting ...
1
vote
1answer
287 views

Function w/ logarithm word problem

Psychologists sometimes use the function $$L(t) = A\left(1 - e^{-kt}\right)$$ to measure the amount $L$ learned at time $t$. The number $A$ represents the amount to be learned, and the ...
0
votes
2answers
100 views

Another induction problem (concrete case)

We have the succession and its formula: $$ 1^2+4^2+\cdots+ (3k-2)^2 = \dfrac{k(6k^2-3k-1)}{2} $$ Now we need to apply it for $k+1$: $$ 1^2+4^2+\cdots+ (3n-2)^2 +(3(k+1)-2)^2 = \\ ...
4
votes
2answers
54 views

Compute the limit

Consider an equation $$ \tan (x) = \frac{a x}{x^2+b} $$ where $a,b \neq 0$. Plotting $\tan(x)$ and function on the RHS we can see that this equation has infinitely many positive solutions $x_{n}$ ...
6
votes
2answers
415 views

What's wrong with my Mac Grapher?

I hope this question is not that stupid. $y=x^{\frac{-2}{3}}$ is $y=\frac{1}{\sqrt[3]{x^{2}}} $right? But when I type $y=x^{\frac{-2}{3}} $and $y=\frac{1}{\sqrt[3]{x^{2}}}$into Mac Grapher,I got ...
1
vote
1answer
267 views

Arc Length Of Parametric Curve

I attached the problem as a file: Where did the trig functions go? I sifted through the different trig identities and formulas, but couldn't find anything that I could use. What should I do?
2
votes
2answers
3k views

Finding Where A Parametric Curve Intersects Itself

The problem I am working on is to find the where the curve intersects itself, using the parametric equations. These are: $x=t^2-t$ and $y=t^3-3t-1$ For the graph to intersect itself, there must be ...
1
vote
3answers
58 views

Showing that $ e^{2x^2} + 2e^{x^2} + 2 = (e^{x^2} + 1)^2 + 1$

I cannot figure out how the left side is converted to the right side...can someone please explain? I have tried some factoring methods but it doesn't give me the right side. $$ e^{2x^2} + 2e^{x^2} + ...
3
votes
1answer
205 views

How would you write the aspect ratio when height is greater than width?

If you have an image that is sized 200x100 you would say the aspect ratio is "2:1" correct? You can figure that out with width/height correct? What if the size was 100x200? Would you say the aspect ...
1
vote
3answers
71 views

I don't know what operation to preform with “!”

I am currently working on my math homework (the book was optional), and I don't know what to do when I see the "!". I am trying to evaluate a problem with this symbol.
2
votes
1answer
237 views

Net selling price with commission and taxes based on percentage desired

Say I have listed my house for sale for $100,000 and I would like to net 90% of my asking price. I have to pay a 2% commission to the buying agent and that commission will be taxed at 13%. How do I ...