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
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.
4
votes
4answers
102 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 ...
1
vote
2answers
42 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
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
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
1answer
135 views

What is the value of $\csc^2\frac{\pi}{14}+\csc^2\frac{3\pi}{14}+\csc^2\frac{5\pi}{14}$? [duplicate]

How to compute $$S=\csc^2\frac{\pi}{14}+\csc^2\frac{3\pi}{14}+\csc^2\frac{5\pi}{14}$$ I tried to rewrite it in terms of $\sin$ $$ ...
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
4answers
106 views

Solve the equation $(m^2-m-2)x=m^2+4m+3$

Here's how I solve it I think that m is the variable (am I right?). Then $$m^2x-mx-2x-m^2-4m-3=0$$ $$m^2(x-1)-m(x+4)-(2x+3)=0$$ $$D=x^2+8x+16+4(x-1)(2x+3)$$ $$=x^2+8x+16+4(2x^2-2x+3x-3)$$ ...
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
6answers
172 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
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
4answers
2k views

How 55 minutes gained in 60 minutes by hour hand

Hello everyone, This is my first question in math.stackexchange. I want to know how the hour hand of the clock "gains" 55 minutes in 60 minutes. I have read this article-: Wiki Answers ...
3
votes
1answer
74 views
+50

Checking logarithm inequality.

Which one of the following is true. $(a.)\ \log_{17} 298=\log_{19} 375 \quad \quad \quad \quad (b.)\ \log_{17} 298<\log_{19} 375\\ (c.)\ \log_{17} 298>\log_{19} 375 \quad \quad ...
1
vote
1answer
119 views

Calculation of $\bf{Max.}$ value of $\sqrt{10x-x^2}-\sqrt{18x-x^2-77}\;\;\forall x\in \mathbb{R}$

(1) Calculation of Max. and Min. value of $\sqrt{x^2-3x+2}+\sqrt{2+3x-x^2}\;\; \forall x\in \mathbb{R}$ (2) Calculation of Max. value of $\sqrt{10x-x^2}-\sqrt{18x-x^2-77}\;\;\forall x\in \mathbb{R}$ ...
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 ...
1
vote
3answers
70 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 ...
-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?
6
votes
5answers
250 views

Show that $\frac{xy}{z} + \frac{xz}{y} + \frac{yz}{x} \geq x+y+z $ by considering homogeneity

Well, I'm preparing for an undergrad competition that is held in April and because of that I've been trying to solve the inequalities I find on the internet. I found this problem: $$\displaystyle ...
1
vote
4answers
173 views

If real number x and y satisfy $(x+5)^2 +(y-12)^2=14^2$ then find the minimum value of $x^2 +y^2$

Problem : If real number x and y satisfy $(x+5)^2 +(y-12)^2=14^2$ then find the minimum value of $x^2 +y^2$ Please suggest how to proceed on this question... I got this problem from ...
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.
3
votes
9answers
251 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 ...
-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 $\sqrt{x^2+5} + \sqrt{x^2-3}$?
20
votes
12answers
3k views

What do sine, tan, cos actually mean?

I know that $\sin\theta=\frac{y}{r}$ and $\cos\theta=\frac{x}{r}$. My question is: is $\sin$ a function of $\theta$, as in $\sin (\theta$)? If yes, why is there no $\theta$ on the right hand side of ...
2
votes
6answers
147 views

Prove the inequality $|xy|\leq\frac{1}{2}(x^2+y^2)$

How can I prove the inequality $|xy|\leq\frac{1}{2}(x^2+y^2)$ I have tried substitute $x,y$ for numbers, which turns out right, but I don't know how to reason here. Thanks in advance!
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
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
1answer
33 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 ...
16
votes
6answers
511 views

$\log_9 71$ or $\log_8 61$

I am trying to know which one is bigger :$$\log_9 71$$ or $$\log_8 61$$ how can i know without using a calculator ?
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 ...
5
votes
5answers
143 views

$f(x) =ax^6 +bx^5+cx^4+dx^3+ex^2+gx+h $ find f(7)

Problem : $f(x) =ax^6 +bx^5+cx^4+dx^3+ex^2+gx+h$ Given that : $f(1)= 1, f(2) =2 , f(3) = 3, f(4) =4, f(5)=5, f(6) =6$ find $f(7) =?$ My approach: We can put the values of $f(1) = 1$ in the ...
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
2answers
89 views

Understanding Bernoulli's Inequality

Here is the proof, I understand it for the most part, but have some specific questions to help me understand better. In the induction step: Line 2, we multiply $(1+x)$ to balance the comparison so ...
1
vote
0answers
36 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 ...
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 ...
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
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
63 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 ...
54
votes
17answers
23k views

What is the most elegant proof of the Pythagorean theorem? [on hold]

The Pythagorean Theorem is one of the most popular to prove by mathematicians, and there are many proofs available (including one from James Garfield). What's the most elegant proof? My favorite ...
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 $a,b,c,d$.
1
vote
1answer
418 views

Algorithm to find the maximum/minimum of a polynomial without graphing.

For a quadratic equation of the form $y=ax^2+bx+c$ the max/min occurs at $x=-\frac{b}{2a}$. Is there any hard and fast equation like this for polynomials of degree $\geq 4$?. For such polynomials the ...
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 [closed]

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
3answers
118 views

Harder-Than-Seems Inverse of $f(x)=x^3-x-12$?

This may seem simple but I have had long days of frustration with finding the inverse of this: $$f(x)=x^3-x-12.$$ I got this on some homework and it did not ask for the inverse. However I wanted to ...
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}) ...
-2
votes
1answer
52 views

Algebraic Problem regarding Cubes. [closed]

If $a^3 + b^3 + c^3 = 3abc$ where $a \ne b \ne c$, find the value of: $(a + b + c)$
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!