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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-4
votes
2answers
27 views

Basic root numbers question

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

A basic root numbers question

If $\sqrt{x^2+5} - \sqrt{x^2-3} = 2$, then what is $\sqrt{x^2+5} + \sqrt{x^2-3}$?
2
votes
8answers
160 views

How is $x \leq x^2$ false?

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
0answers
26 views

Determining polynomial values

I need to determine the values of $A$ and $B$ in this polynomial equation. $R(x)=x^4+Ax^3+Bx^2+10x-1$ when divided by $x+1$ the remainder is $-15$.
0
votes
1answer
14 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
3answers
26 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
25 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
2answers
38 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
73 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 ...
0
votes
1answer
44 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
23 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 ...
1
vote
2answers
26 views

Remainder Dividing Repunits

If n= 11111....1 ( 1 repeated 123 times.) Then find the remainder when n is divided by 271 ? I know I can write this in the form of a sum of a gp but it doesn't help to find the remainder... Any ...
0
votes
1answer
23 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
57 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
63 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
46 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
41 views

Finding the intersection of a line and standard wave function

Let's say I have two functions: $f(x)=5cos(x)$ $f(x)=4$ 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
57 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
72 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
57 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
29 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
61 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
46 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
47 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
47 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
39 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
51 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
42 views

Algebra 2 help! [on hold]

Write the equation of a circle with the given center and radius. Center (-2,3); Radius 8
1
vote
2answers
33 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
109 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
40 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
28 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
53 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
56 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
63 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 $x=1, -2$
1
vote
1answer
33 views

Find all ordered triplets $(p,q,r)$

Let $A,G^2,H$ be the roots of the cubic equation $x^3+px^2+qx+r=0$ which are in G.P., where $p,q$ are integers and $A,G,H$ are respectively AM,GM,HM of two positive numbers. If $p,q \in (-100,100)$, ...
0
votes
0answers
37 views

Why in order to have the greatest term in the expansion of$ (1 + x)^n$, $x$ can't be greater than unity?

I was reading Binomial theorem of any index of Higher Algebra by Hall & Knight; there a section was attributed to find the greatest term in the expansion of $(1+x)^n$ for any rational value of ...
0
votes
2answers
57 views

For which $x, y\in\mathbb{R ^+}$ do we have $|xy-\frac{1}{xy}|\le|x-\frac{1}{x}|+|y-\frac{1}{y}|$?

I need to find all $x, y\in\mathbb{R^+}$ such that the following inequality holds. $$\Big| xy-\dfrac{1}{xy}\Big|\le\Big|x-\dfrac{1}{x}\Big|+\Big|y-\dfrac{1}{y}\Big|$$ If I substitute $x=2$ and $y=3$ ...
2
votes
6answers
58 views

Why does $|x_1| = |x_2| \implies x_1 = \pm x_2$

I was doing a 'prove this is not surjective' practice problem and the step leading from my hypothesis, as listed, to the conclusion was not defined. I don't recall being exposed to a situation where ...
-2
votes
1answer
39 views

How would you divide a polynomial by another polynomial whose power is greater than its nominator? [on hold]

I have a polynomial which is: $$\frac{(x^3-4x)}{(4x^2-4x+1)} = -10$$ Is there a way to do this? I have thought about doing long division which was not helpful...
3
votes
3answers
159 views

Definite integral with limits from zero to infinity

Let $ I=\int\limits_{0}^{\infty}e^{-(x^2+\frac{1}{x^2})}dx$ and $J=\int\limits_{0}^{\infty}x^2e^{-(x^2+\frac{1}{x^2})}dx$. If $J=\dfrac{pI}{q}$, then find the value of $p+q$ where $p$ and $q$ are ...
0
votes
2answers
34 views

Finding the correct slope.

To determine the slope of the graph of this relation do I take the two points as (4,20), (0,0) and then proceed to take 20-0=20 and 4-0=4, to divide 20 by 4 to get the slope of 5m? For the ...
-2
votes
3answers
117 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 ...
1
vote
2answers
39 views

How do I interpret this question: Do I multiply, divide, subtract first?

Which of the following expresses $6p+2py-4p$ in its simplest form? (A) $2p+2py$ (B) $4py$ (C) $4p^3y$ (D) $10p+2py$ Im not really sure how to go about it...
6
votes
5answers
111 views

Calculate the limit $\lim_{x\to 0} \left(\dfrac 1{x^2}-\cot^2x\right)$ [on hold]

The answer of the given limit is $2/3$, but I cannot reach it. I have tried to use the L'Hospital rule, but I couldn't drive it to the end. Please give a detailed solution! $$\lim_{x\to 0} ...
0
votes
4answers
49 views

Lawn mowing problem solving

Kate can mow the lawn in 45 minutes. Kate's sister takes twice as long to mow the same lawn. If they both have a mower and mow the lawn together, how many minutes will it take them? I know the answer ...