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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0
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2answers
121 views

cost price of item

let us consider following problem: Roger sold a watch at a profit of $10$%. If he had bought it at $10\%$ less and sold it for $13$ dollar less,then he would have made a profit of $15$%. What is ...
1
vote
2answers
117 views

crossing path of two rail

let us consider following problem Two trains starting from cities $300$ miles apart head in opposite directions at rates of $70$ mph and $50$ mph, respectively. How long does it take the trains to ...
8
votes
2answers
85 views

Why can we use inspection for solving equation with multiple unknowns?

In our algebra class, our teacher often does the following: $a + b\sqrt{2} = 5 + 3\sqrt{2} \implies \;\text{(by inspection)}\; a=5, b = 3 $ I asked her why we can make this statement. She was ...
3
votes
1answer
63 views

Existance and uniqueness of solution for a point with fixed distances to three other points

I have two sets of known points in $\mathbb{R}^2$: Four points $\mathbf{p_1}, \mathbf{p_2}, \mathbf{p_3}, \mathbf{p_x}$ , and three other points $\mathbf{q_1}, \mathbf{q_2}, \mathbf{q_3}$. I would ...
0
votes
1answer
62 views

Solving log/exponent equation

I've transformed the number 11 to: $11^e = 677.32$ Given the exponent and the transformed value, how can I solve for the original number? I know that $x = y^z$ and that $z = \log_y(x)$, but I don't ...
26
votes
2answers
2k views

Factorize $(x+1)(x+2)(x+3)(x+6)- 3x^2$

I'm preparing for an exam and was solving a few sample questions when I got this question - Factorize : $$(x+1)(x+2)(x+3)(x+6)- 3x^2$$ I don't really know where to start, but I expanded everything to ...
0
votes
1answer
127 views

How are boolean expressions converted to NOR expressions?

What kind of rules help to convert an expression into a 3 input NOR expression? Do all variables have to be of the form (a+b+c)' + (d+e+f)'? What happens if there is an expression that is just (a')' ...
0
votes
2answers
53 views

What i am doing wrong here . and if i am correct then can any one explain to me how $1=-1$ [duplicate]

Let suppose $(-1)(-1)=(-1)²$ if i take square root on both sides then $\sqrt{(-1)(-1)}=\sqrt{(-1)²}$ after multiplication and cancelation $\sqrt{1}=(-1)$ which is wrong although i suppose a ...
6
votes
2answers
135 views

How to find the sum $\displaystyle\sum^n_{k=1} (k^2+k+1)k!$

How to find the sum $$\sum^n_{k=1} (k^2+k+1)k!$$ What I tried as follows : $$\sum^n_{k=1} (k^2+k+1)k!$$ =$$\sum^n_{k=1} (k^2)k!+ \sum^n_{k=1} (k)k! + \sum^n_{k=1} (1)k!$$ Now we can write ...
1
vote
1answer
228 views

Solving inequalities at powers greater than $2$

Usually when I see an inequality like $x^2 - 6x - 16 < 0$, I know that the answer is $-2 < x < 8$ because I can picture where the graph would lie below zero. However, for a problem like ...
2
votes
3answers
305 views

Show that $\gcd(a + b, a^2 + b^2) = 1\mbox{ or } 2$ [duplicate]

How to show that $\gcd(a + b, a^2 + b^2) = 1\mbox{ or } 2$ for coprime $a$ and $b$? I know the fact that $\gcd(a,b)=1$ implies $\gcd(a,b^2)=1$ and $\gcd(a^2,b)=1$, but how do I apply this to that?
2
votes
2answers
498 views

Inverse of a function $e^x + 2e^{2x}$

The function is $f(x) = e^x+2e^{2x}$ So to find the inverse I went $$WTS: x = ____$$ $$y = e^x+2e^{2x}$$ $$ log3(y)=log3(3e^{2x})$$ $$ log3(y) = 2x$$ $$ log3(y)=5x$$ $$ x=\frac{log3(y)}{2}$$ Am ...
0
votes
1answer
249 views

Write the indicated term of the binomial expansion [closed]

$(5x+5)^5$; 5th term Please be specific. and show work
0
votes
2answers
89 views

Prove $|x+1|\leq 4$ implies that $-4\leq x\leq 2$.

How do I prove that if $x$ is a real number, then $\lvert x+1 \rvert\leq 3$ implies that $-4\leq x\leq 2$. EDIT: $\lvert x+1 \rvert\leq 4$ should be $\lvert x+1 \rvert\leq 3$
2
votes
3answers
98 views

Does the graph of $y + |y| = x + |x|$ represent a function of $x$?

The question is whether or not the graph $y + |y| = x + |x|$ represents a function of $x$. Explain why. It looks like a weird graph but it would probably be a function because if you say $f(x) = y$ ...
0
votes
1answer
200 views

Is it possible to prove $|\sin(x)| \leq 1$, $|\cos(x)| \leq 1$ and $|\sin(x)| \leq |x|$ algebraically?

I know that we can prove $\forall x \in \mathbb{R}: |\sin(x)| \leq |x|$ by using the mean value theorem on $\sin(x)$, it's also easy to see that when $x \in \mathbb{R}$ we have $|\sin(x)| \leq 1$, ...
12
votes
1answer
373 views

When $\sin x, \cos x$ are $\mathbb{Q}$-linear combinations of square roots

Suppose $x\in\Bbb R$ is such that $$\sin x=\sum_{i=1}^m x_i\sqrt{r_i},\quad \cos x=\sum_{j=1}^n y_j\sqrt{s_j}$$ for some $x_i, r_i, y_j, s_j \in\Bbb Q \ , \ |x_i|=|y_j|=1$. Show that ...
-1
votes
1answer
44 views

Show that $|uv|^2 = s+t$ when $(a^2+b^2)(c^2+d^2)=s^2+t^2$

$$\text{Show that }|uv|^2 = s+t\text{ when }(a^2+b^2)(c^2+d^2)=s^2+t^2.$$ $$\text{ALSO } u = a+bi, v=c+di...(a,b,c,d)\text{ are integers.}$$ So using this somehow I have to show that 17$\times$29 = ...
0
votes
1answer
337 views

Calculating z/w in polar form

IT HAS BEEN ANSWERED, THANK YOU So we have: z = 7 + 4i w = 8-i I did this: $$7+4i/8-i$$ $$conjagate =\space 8+i$$ $$ (7+4i/8-i) \times (8+i/8+i)$$ $$(56+7i+32i+4i^2)/((8-i)\times(8+i))$$ ...
2
votes
3answers
308 views

Find the area of cd without a diameter or radius

A compact disc (CD) is made such that the shortest distance between the edge of the centre hole and the edge of the disc is 53.0 mm. Find the radius of the centre-hole if 1.36% of the disc is removed ...
2
votes
3answers
96 views

How to solve this : $ \prod^{\infty}_{n=2}(1-\frac{1}{n^2}) $

How to solve this : $$ \prod^{\infty}_{n=2}(1-\frac{1}{n^2}) $$ We can write it as follows : $$ \prod^{\infty}_{n=2}\frac{(n-1)(n+1)}{n^2} $$ How can we proceed further please guide.. thanks.. ...
1
vote
1answer
71 views

How to find Find $\left\lfloor\sum_1^{100}\frac{1}{x_n+1}\right\rfloor$ with $x_1 =\frac{1}{2}, x_{k+1} =x_k^2+x_k$.

The sequence $\{x_n\}$ is defined by $x_1 =\frac{1}{2}, x_{k+1} =x_k^2+x_k$. Find $$\left\lfloor\frac{1}{x_1+1}+\frac{1}{x_2+1}+\cdots+\frac{1}{x_{100}+1}\right\rfloor$$ where ...
3
votes
1answer
214 views

Where is the mistake in this argument that $(\sqrt8)^{\sqrt 7} >(\sqrt7)^{\sqrt 8}$?

I posted an answer in this question to prove that $(\sqrt8)^{\sqrt 7}<(\sqrt7)^{\sqrt 8}$ I started with $$f(x)=\frac{\ln x}{x}$$ $$f'(x)=\frac{1-\ln x}{x^2}$$ $$f'(x)>0 : x\in)0,e($$ ...
2
votes
1answer
75 views

quantity comparative test on GRE

i would like to compare two quantity,first is $-3^4$ and $(-3)^4$,second one is clearly $81$,but what about second on?is it $-3\cdot-3\cdot-3\cdot-3$ ? $-3\cdot3\cdot3\cdot3$?there could be ...
2
votes
3answers
77 views

show that $n^2 + (n+1)^2 = 2m^2$ is impossible

show that $n^2 + (n+1)^2 = 2m^2$ is impossible If $m=n$, $n^2 + (n+1)^2$ does not equal $2n^2$ But if $m$ does not equal $n$, how do you prove this?
8
votes
4answers
321 views

Prove the inequality $\,\frac{1}{\sqrt{1}+ \sqrt{3}} +\frac{1}{\sqrt{5}+ \sqrt{7} }+\ldots+\frac{1}{\sqrt{9997}+\sqrt{9999}}\gt 24$

Prove the inequality $$\frac{1}{\sqrt{1}+ \sqrt{3}} +\frac{1}{\sqrt{5}+ \sqrt{7} }+......... +\frac{1}{\sqrt{9997}+\sqrt{9999}} > 24$$ My work: Rationalizing the denominator gives ...
5
votes
3answers
278 views

How to solve this : $\prod^{\infty}_{n=2} \frac{n^3-1}{n^3+1}$

How to find the sum of this : $$\prod^{\infty}_{n=2} \frac{n^3-1}{n^3+1}$$ My Working : $$\prod^{\infty}_{n=2} \frac{n^3-1}{n^3+1}= 1 - \prod^{\infty}_{n=2}\frac{2}{n^3+1} = 1-0 = 1$$ Is it ...
0
votes
1answer
117 views

How to solve : $\sum^{\infty}_{k=1} \frac{6^k}{(3^k-2^k)(3^{k+1}-2^{k+1})}$

How to find the sum of this : $$ \sum^{\infty}_{k=1} \dfrac{6^k}{(3^k-2^k)(3^{k+1}-2^{k+1})} $$ I tried to find the partial fraction of this but I think this is wrong method ...please suggest how to ...
1
vote
1answer
68 views

If n=(sin^2(2x))/4cos^2(x))+1/(sec^2(x)) and x=2.01307, find 2013n^2013

If $n=\dfrac{sin^2(2x)}{4cos^2(x)+\dfrac{1}{sec^2(x)}}$ and $x=2.01307$, find 2013n^2013 Your edits are wrong! These are two separate fractions not together!anymore!
9
votes
3answers
696 views
1
vote
2answers
129 views

Solving “Real Life” Problems

Michelle earns 5 dollars an hour when she works up to 10 hours a day. Michelle earns 12 dollars an hour for each hour over 10 hours a day she works. If she earned 92 dollars on Tuesday, how many hours ...
4
votes
1answer
82 views

Need proofread for deriving quadratic equation formula

How to Solve quadratic equation $$ax^{2}+bx+c=0$$ such as $$a \neq 0$$ Divide by a both side from the equation such as $$\frac{a}{a}x^2 + \frac{b}{a}x + \frac{c}{a} = 0$$ $$\Rightarrow ...
5
votes
3answers
222 views

Starting with $\frac{-1}{1}=\frac{1}{-1}$ and taking square root: proves $1=-1$

In this blog post, RJ. Lipton mentions an example of common mathematical traps. In particular, that ``square root is not a function''. He shows the following trap: Start with: ...
1
vote
3answers
80 views

How to go from $5\pi/4$ to $\pi + \pi/4$?

Use standard triangles to find exact value of $\cos(5\pi/4)$. Example states that $5\pi/4$ is equal to $\pi + \pi/4$ but doesn't list the steps to get $\pi + \pi/4$...
2
votes
2answers
116 views

2 limits logarithms $ \lim_{t\to \infty} t-t^2\ln(\frac{t+1}{t}) $

I have problems with the following limits logarithms, and I can not use L'Hopital or power series (I know the results of the problems, using these methods), so I need solutions that do not occupy ...
0
votes
1answer
34 views

Applying down-payments

I have to ask customers for 1/3 down from their total bill before work can be performed. How do I calculate this?! For example, if a total bill was $688.96, how do I calculate to show their down ...
2
votes
4answers
206 views

a dog tied to a pole by a rope

A square hole of depth $h$ whose base is of length $a$ is given. A dog is tied to the center of the square at the bottom of the hole by a rope of length $L>\sqrt{2a^2+h^2}$ ,and walks on the ...
4
votes
1answer
138 views

Logic arguments applied to a false math statement

I was asked to demonstrate the next equality: $$1^2 + 2^2 + 3^2 + ... + n^2=\frac{(n(n+1)(n+2))}6$$ Now I am trying to express correctly what kind of error appears in the statement. My questions ...
0
votes
2answers
83 views

Solve combinatorics problem to target desired result

I wonder if someone would be able to suggest some solution, programming technique or, at least, the right name for the problem, so I could research more. I have a problem where I given a number of ...
2
votes
1answer
109 views

Quadratic inequality with boundaries

Here is a very old high school exam question I am trying to solve (purely for interest only): If $a,b,c$ are real numbers such that $-1 \le ax^2+bx+c \le 1$ for $-1 \le x \le 1$ prove that $-4 \le ...
1
vote
2answers
71 views

Question about the imaginary unit [duplicate]

As we know, we define $$\sqrt{-1}=i$$ But I always wondered, what about $\sqrt{i}$? As far as i can see, it is not an integer power of $i$. Every odd root has a solution in an integer power of $i$, ...
0
votes
1answer
84 views

How to have the best solution for this equation $4\sqrt{x^5}-10x^2-(5-8\sqrt{x})x+\sqrt{x}=0$

How to have the best solution for this equation $4\sqrt{x^5}-10x^2-(5-8\sqrt{x})x+\sqrt{x}=0$ (1) I set $t=\sqrt{x}$ and $(1) \Leftrightarrow t(4t^4-10t^3-5t^2+8t+1)=0$ So It's difficult :( Thank ...
0
votes
1answer
126 views

Determine all the values of the parameter $a$ for which the inequality $3-|x-a|>x^2$ is satisfied by at least one negative $x$.

I wanted to know, how can I determine all the values of the parameter $a$ for which the inequality $3 - |x-a| > x^2$ is satisfied by at least one negative $x$. I tried for $x<a, |x-a|=-(x-a)$ ...
67
votes
13answers
5k views

What would have been our number system if humans had more than 10 fingers? Try to solve this puzzle.

Try to solve this puzzle: The first expedition to Mars found only the ruins of a civilization. From the artifacts and pictures, the explorers deduced that the creatures who produced this ...
0
votes
1answer
156 views

Arctanh to exp: Prove two equations are equivalent

For some peace of mind in a project, I am trying to prove two equations are somewhat equivalent. I have these two equations. $$ i_{1} = ...
1
vote
4answers
111 views

How to find the value of $a$ for which $\;\tan^2x + (a+1)\tan x-(a-3)<0$ is true

I wanted to know, how can I find the value of $a$ for which the inequality $\tan^2x + (a+1)\tan x-(a-3)<0$ is true for at least one $x\in(0,\pi/2)$. I don't know how to proceed, any help is ...
0
votes
2answers
439 views

Maximum number of cans within a box

A box that is 4 ft. by 4 ft. by 4 ft. is packed with (cylindrical) cans that are 2 ft. high and have a diameter of 6 inches. When the box is fully packed with cans, how much space is wasted in the ...
3
votes
2answers
92 views

Question on radius problem

A car with 15 inch radius tires was driven on a trip of a distance equal to 400 miles. Two months later, with snow tires, the odometer indicated 390 miles for the same trip. Find the radius of the ...
0
votes
1answer
73 views

Exponentials, Logarithms & the Natural Log

Could someone show me how to solve this problem? I don't know what "\" in front of "ln" means either. Solve the following equation for $y = f(x)$. $$e^y = e^{2y}e^{\ln(2x)}$$
0
votes
1answer
650 views

Folding a paper in half - Crease Lines

A strip of paper is folded in half, then the result is folded in half again, and the process is repeated for a total of 6 times (including the first fold). How many creases (fold lines) are there?