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.

learn more… | top users | synonyms (2)

1
vote
2answers
41 views

Show $n \cdot \log_b r + \log_b \frac{r}{r-1} \le \lceil n \cdot \log_b r \rceil$

Let $n$ be a natural number and $b, r > 1$ be two natural numbers, then I guess we have $$ n \cdot \log_b r + \log_b \frac{r}{r-1} \le \lceil n \cdot \log_b r \rceil. $$ where $\lceil x \rceil = ...
4
votes
4answers
99 views

Greatest of the numbers given [duplicate]

To find out the greatest among the number given below: $3^{1/3}, 2^{1/2}, 6^{1/6}, 1, 7^{1/7}$ I have plotted the following graph using graph plotter which is shown below: It can be concluded that ...
-4
votes
0answers
31 views

finding roots of polynomial equation [on hold]

the product of two roots of the equation 4x^2-24x^3+31x^2+6x-8=0 is 1, find all the roots
1
vote
1answer
36 views

Show that $x^2+y^2$ is constant for all values of $\theta$.

Given that $x=3\sin \theta-2 \cos \theta$ and $y=3\cos \theta+2 \sin \theta$ i)Find the value of the acute angle $\theta$ for which $x=y$ ii)Show that $x^2+y^2$ is constant for all values of ...
1
vote
1answer
27 views

Express various trig functions in terms of the sine.

The acute angle $x$ radians is such that $\sin x = k$, where $k$ is a positive constant. Express, in terms of $k$. i) $\sin (2\pi-x)$ ii) $\tan(\frac{1}{2}\pi-x)$ iii) $\cos (\pi+x)$ My attempt: ...
1
vote
2answers
79 views

Is Spivak wrong here, or am I just missing something?

Chapter 1 Problem 18 has the reader doing various proofs with second-degree polynomial functions of the form $x^2 + bx + c$. My issue lies with problem 18d, but it uses knowledge from 18b and 18c, so ...
0
votes
1answer
67 views

algebra question.. [on hold]

If $f : \mathbb{R}\rightarrow \mathbb{R}$, and $f(x)=\frac{2}{4^{x}+2}$ Find the value of $$f\left [ \frac{1}{11} \right ]+f\left [ \frac{2}{11} \right ]+ \cdots +f\left [ \frac{10}{11} \right ]$$
1
vote
1answer
21 views

Sorting triangles by hypotenuse length

I have some points in $xy$ space and I need to sort distances between these points. If I calculate real distance, then I need to perform $\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}$ and this is very time ...
1
vote
6answers
171 views

Why is $\frac{1}{4/3} - \frac{1}{3/2}$ not the same as $\bigl(\frac{4}{3} - \frac{3}{2}\bigr)^{-1}$

If you have the problem:$$\frac{1}{\frac{4}{3}} - \frac{1}{\frac{3}{2}} =?$$ Why can't you change the problem to $(\frac{4}{3} - \frac{3}{2})^{-1}$ and get the same answer? In the first scenario, ...
-1
votes
2answers
27 views

Compound interest 10% per 10 seconds [on hold]

We are starting with 354, ending with 700'000. The interest is 10% every 10 seconds. How long will it take to reach the final figure?
2
votes
3answers
69 views

find the complex number $z^4$

Let $z = a + bi$ be the complex number with $|z| = 5$ and $b > 0$ such that the distance between $(1 + 2i)z^3$ and $z^5$ is maximized, and let $z^4 = c + di$. Find $c+d$. I got that the ...
2
votes
3answers
42 views

Proving $|x+y|=|x|+|y| \iff x\cdot y \geq 0$

Prove: $|x+y|=|x|+|y| \iff x\cdot y \geq 0$. $|x+y|=|x|+|y| \iff x+y=x+y$ and $-(x+y)=-x-y \iff \{x,y\}\geq 0$ and $\{x,y\}\leq 0 \iff x\cdot y\geq 0$ in both cases.
-4
votes
2answers
29 views

Basic root numbers question [on hold]

Hello I want to simplify this expression $1\over\sqrt{(2-\sqrt{5})^2}$ Thank you
1
vote
4answers
136 views

A basic root numbers question

If $\sqrt{x^2+5} - \sqrt{x^2-3} = 2$, then what is $\sqrt{x^2+5} + \sqrt{x^2-3}$?
3
votes
9answers
248 views

How is $x \leq x^2$ false? [on hold]

There's an equation that says $$x \leq x^2$$ and $x \in \mathbb R$. What I can solve and clearly see is that this equation would be true for any value of '$x$' but then how come my maths teacher ...
0
votes
1answer
60 views

Determining polynomial values

The polynomial has been edited to include the "x" term $R(x)= x^4+Ax^3+Bx^2+10x-1$ has a remainder of $-15$ when divided by $x+1$ and a remainder of $39$ when divided by $x-2$. Determine $A$ and ...
0
votes
1answer
17 views

Quadratic equation roots values was positive but shown as negative

Hi, This screen capture was taken from KhanAcademy. I am an adult learner trying to revisit Algebra I/II concepts. In the video, p was calculated as 1/4 or 4. But, why was is factorized as ...
0
votes
2answers
32 views

Find the number of seven digit whole numbers in which only 2 and 3 are present as digits if no two 2's are consecutive in any number?

Find the number of seven digit whole numbers in which only $2$ and $3$ are present as digits if no two $2$'s are consecutive in any number? My Approach: We can make numbers and see like: ...
0
votes
1answer
31 views

If $500! = 2^m\cdot$N, where N is an odd positive integer, then find $m$

Problem : If $500! = 2^m\cdot$N, where N is an odd positive integer, then find $m$ My approach : Shall we need to expand $500!$ and then find prime factors and see what is the power of 2 in that ...
-1
votes
3answers
61 views

Find all solutions to the equation. $7 \sin^2x - 14 \sin x + 2 = -5$

I got this question wrong on a test and I want to see what I did wrong so I don't get this type of question wrong again.
2
votes
3answers
81 views

In what conditions a quadratic function has an integer value of $f(x)$ where $x$ is also an integer?

EDITED Sorry, the question was wrong. Please forgive me for this. Suppose a quadratic function $f(x) = ax^2+bx+c$, what I want to know is if in an integer $x$, say $x=1, x=2, x=3, ...$, the function ...
1
vote
2answers
68 views

Vibrating water container problem

I am struggling with this seemingly difficult question: "A water-filled container is sitting still on a platform as shown. Suddenly, the platform starts shaking vertically due to the action of a ...
1
vote
0answers
34 views

Proving an inequality involving discrete variables

I'm trying to show that the following inequality holds $$ \frac{1-x^{n}}{1-x^{n+1}}\geq\frac{\sum_{i=0}^{n-2}x^{i}(1-x_{1}^{n-(i+1)})}{\sum_{i=0}^{n-1}x^{i}(1-x_{1}^{n-i})}, $$ where $n$ is a ...
0
votes
1answer
29 views

3 Points in 3D Space to Develop an Arc or Circle

Background: I'm a Robotics Engineer and I am trying to develop a more flexible, modular, and robust program for our welding robots, which will minimize teaching time for new robots and also minimize ...
1
vote
0answers
62 views

Find the number of “p-safe numbers”

For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$. For example, the set of $10$-safe numbers is ...
0
votes
2answers
67 views

Find all the possible real values for $a,b,c,d$.

Let pairs $(a,c)$ and $(b,d)$ be roots of the equations $x^2 + ax - b = 0$ and $x^2 + cx + d = 0$ respectively. Find all possible real values for $a,b,c,d$.
2
votes
1answer
47 views

Find a constant $C$ such that $ \Bigg| \frac{\prod_{i=0}^{k-1} (n-i) }{n^k} - 1 \Bigg| \leq \frac{C}{n}, \forall k \leq n$

Consider the following: $$ \Bigg| \dfrac{\prod_{i=0}^{k-1} (n-i) }{n^k} - 1 \Bigg| \leq \frac{C}{n}, \forall k \leq n $$ How to find an expression for $C$ independent of $k$ and thus $n$? It arises ...
1
vote
2answers
47 views

Finding the intersection of a line and standard wave function

Let's say I have two functions $f(x)=5\cos(x)$ and $g(x)=4$. The line lies in between the range of the wave function so there will be two intersections for each period of the wave. I want to find ...
3
votes
2answers
58 views

Finding the equation of a circle.

A circle of radius $2$ lies in the first quadrant touching both axis. Find the equation of the circle centered at $(6,5)$ and touching the above circle externally. Let me share how I answered this ...
2
votes
5answers
73 views

Find the equation of the circle.

Find the equation of the circle whose radius is $5$ which touches the circle $x^2 + y^2 - 2x -4y - 20 = 0$ externally at the point $(5,5)$
0
votes
1answer
18 views

equation solving approach for the given equation

$y^4-10y^3+5y^2+100y+100=0$ how to solve this equation ? what will be the approach to solve this question. Breaking $100$ into $84$ and $16$ and then solving could be one .
1
vote
2answers
58 views

This expression is always a perfect square [on hold]

How to show that for $x,y\in \Bbb R$, the expression $xy+\left(\frac{x-y}{2} \right)^2$ is always a perfect square? For example $x=7, y=3$, $7\times 3+\left(\frac{7-3}{2} \right)^2=25=5^2$
2
votes
2answers
31 views

Parallelogram ABCD

There's a parallelogram $ABCD$. I'm given point $A(3,12)$ and point $B(-1,5)$. Given the equations of the lines $BC$ and $AC$ are $y=8x+13$ and $y=3x+3$ respectively. How to find the coordinates of ...
1
vote
1answer
64 views

How to find x,y,z such that $\frac{1800}{x}=a$, $\frac{1800}{y}=b$, $\frac{1800}{z}=c$, $\frac{1800}{a+b+c}=d$

I'm really fascinated by how questions and problems are designed in mathematics. So, I was designing a simple word problem, and in the course I fell into this situation: a,b,c,d are natural numbers. ...
2
votes
0answers
52 views

Is there a closed-form solution (even approximated) to this inequality?

I have the following function: $f(x, \theta) = (1-\theta)(x+1)^{-\theta}\left[ \frac{2-2\theta}{1- 2\theta} (N^{1-2\theta} - (x+1)^{1-2\theta}) - (x+1)^{-\theta}(N^{1-\theta} - (x+1)^{1-\theta}) ...
1
vote
2answers
48 views

Intuition: inverse function

Disclaimer: I'm a beginner with inverse functions. Can anyone explain what I'm doing wrong here? I'd like to avoid using "y" -- that is, I want to show everything in terms of x and f(x). Thanks!
0
votes
3answers
51 views

If a given # is $70$% of $X$. How do you determine what $X$ is? [duplicate]

Given I have the number $50,000$ which is $70$% of $X$. How do I calculate what $X$ is without guessing. Thanks
-1
votes
1answer
12 views

How to find the bearing and velocity of a boat on a flowing river

Point A is on the west bank of a river and point C is directly across from it on the east bank. The river is 648.6 meters wide and flows south at 2.45 km/hr. A boat wants to cross the river from point ...
1
vote
1answer
40 views

How to solve a equation task

I have found a equation task that is: y=0,5x+2,5 y=3x-1 The answer and its process is: ...
-2
votes
1answer
52 views

Algebraic Problem regarding Cubes. [on hold]

If $a^3 + b^3 + c^3 = 3abc$ where $a \ne b \ne c$, find the value of: $(a + b + c)$
-9
votes
1answer
43 views

Algebra 2 help! [closed]

Write the equation of a circle with the given center and radius. Center (-2,3); Radius 8
1
vote
2answers
35 views

How to show that this interesting difference of products is $O \left( \frac{1}{n^2} \right) $

Let $k \leq n$. Consider the following difference of products: $$ \prod_{i=1}^{k-1} \left( 1 - \frac{i}{n+1} \right) - \prod_{i=1}^{k-1} \left( 1 - \frac{i}{n} \right)$$ For $n=1,2,3$, this is ...
4
votes
4answers
113 views

Is $\left(45+29\sqrt{2}\right)^{1/3} + \left(45-29\sqrt{2}\right)^{1/3}$ an integer?

The problem is the following: Prove that this number $$x = \left(45+29\sqrt{2}\right)^{1/3} + \left(45-29\sqrt{2}\right)^{1/3}$$ is an integer. Show which integer it is. I thought that it ...
0
votes
0answers
43 views

Can you verify the combinatoric recurrence?

There are $2^{10} = 1024$ possible 10-letter strings in which each letter is either an A or a B. Find the number of such strings that do not have more than 3 adjacent letters that are identical. ...
0
votes
1answer
24 views

How to prove that $(a-b) \mod N = a \mod N + ((-b) \mod N)$?

I've gone through the similar post Modulo of a negative number . But that post is not about proof and I'm asking for the proof in general. This question is another follow up question of my previous ...
0
votes
1answer
31 views

Question about the polynomial remainder theorem

Given that $f(x) = x^3 - x^2 - ax - b$ has a factor $x - 3$ but leaves a remainder of $13x - 11$ when divided by $x + 4$, find $a$ and $b$. I get that in order for $x-3$ to be a factor then according ...
1
vote
1answer
57 views

AMT - Three whole numbers add up to 149 and multiply to give 987. What is the largest of the three number

So about this question I'm not too sure... Can't find out what I should start off with. If anyone can help me I'll be very greatly appreciated. The question is: Three whole numbers add up to 149 ...
-3
votes
1answer
57 views

Help me understand probability..I don't get it!! Exam tomorrow

Postcodes can be made from 4 digits --> 1234 How many different postcodes beginning with 2 are possible? How do I truly understand probability questions? Nothing beyond the level of ordered / ...
1
vote
3answers
64 views

how can i prove this trigonometry equation

I need help on proving the following: $$\frac{\cos {7x} - \cos {x} + \sin {3x}}{ \sin {7x} + \sin {x} - \cos {3x} }= -\tan {3x}$$ So far I've only gotten to this step: $$\frac{-2 \sin {4x} \sin {3x} ...
3
votes
4answers
76 views

Solve for $x$ - Logarithm Equation $\ln x+\ln(x+1)=\ln 2$

My attempt: $\ln x(x+1)=\ln 2$ $e^{\ln x(x+1)}=e^{\ln 2}$ $x(x+1)=2$ $x^2+x-2=0$ $(x-1)(x+2)=0$ therefore $x=1, -2$