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
0answers
36 views

Algebra fraction simplification? [closed]

I have the following and is solution but I don't understand how they arrived at the solution.Can someone direct me to how they arrived at this please ? $$\frac{z(z-1)(z+1)}{(z-3)^3}$$ The solution : ...
0
votes
2answers
36 views

Can a equation be represented by more than one graph of different dimensions

Consider the equation $x^2 + y^2 - 0z = 1$ If you only consider the x-y plane, this will trace out the well-known unit circle. However, if you consider 3d space, a cylinder of infinite height without ...
3
votes
1answer
42 views

Finding all possible pairs of positive integer values

The ratio of the sum of two positive integers to their difference is $7:5$. If the the sum of the two numbers is at most $25$, find all possible values for the pair of numbers. Let $m$ be the first ...
0
votes
1answer
30 views

Are these two definitions of an affine subspace equivalent?

I've seen the notion of an affine subspace defined differently as follows: $S \subset \mathbb R^3$, non-empty, is an affine subspace if $(1-t)u + tv \in S$ whenever $u,v \in S$. $S$ is an affine ...
-2
votes
0answers
28 views

Calendar question [closed]

A certain day, which is $x$ days before $17$th of August, is such that $50$ days prior to that day, it was $4x$ days since March $40$th of the same year. What is $x$?
-4
votes
1answer
71 views

Do you agree with this!

So yeah, I stuck with this theory. I thought -4 Square = 16
0
votes
1answer
22 views

Simple Harmonic Motion Formula

The SHM general formula is this: $y(t) = A\sin(\omega t + \alpha) +B$ I have two questions about it As far as I know, there is the cosine formula for when the particle starts at P and sine for ...
0
votes
3answers
49 views

Simple interest problem

I'm new to this site. I'm not sure exactly what appropriate tag I should use, but if anyone could help me with the tags, thens thats great :-) Anyways, I'm having difficulty figuring out a solution to ...
4
votes
2answers
112 views

Evaluate $\lim_{n \to \infty} \int_{0}^1 \frac{n+1}{2^{n+1}} \left(\frac{(t+1)^{n+1}-(1-t)^{n+1}}{t}\right) \mathrm{d}t$

Evaluate $$\lim_{n \to \infty} \int_{0}^1 \frac{n+1}{2^{n+1}} \left(\frac{(t+1)^{n+1}-(1-t)^{n+1}}{t}\right) \mathrm{d}t$$ For this integral, I have tried using integration by parts and then ...
-1
votes
1answer
43 views

How can I prove by mathematical induction that $\sum_{i=0}^n i^4 = (\sum_{i=0}^n i)^3$?

How can I prove by mathematical induction that $$\sum_{i=0}^n i^4 = (\sum_{i=0}^n i)^3$$ ? I see easily that it holds for $i=0$. Using the inductive hypothesis, I get: $$\sum_{i=0}^{n+1} i^4 = ...
3
votes
4answers
59 views

$a^2b + abc + a^2c + ac^2 + b^2a + b^2c + abc +bc^2$ factorisation

I came across this from a university mathematics resource page but they do not provide answer to this. What I did was this: $(a^2+b^2+c^2)(a+b+c) - (a^3 + b^3 + c^3) + 2abc$ But I don't think ...
1
vote
1answer
18 views

Assign set in probability generating functional

say i have an integral $A = \lambda c_d \displaystyle\int_0^\sim (1-exp(-shr^{-\alpha}))dr^{d-1}\mathrm {d}r$ $A = \lambda c_d \displaystyle\int_0^\sim (1-exp(-shr^{-1/\delta}))\mathrm {d}r$ (subs. r ...
1
vote
1answer
44 views

I need to help to figure out a solution to this problem [closed]

Jennifer drove at steady speed for $2$ hours, then reduced speed by $10$mph and drove for another $3$ hours. What was the initial speed if the total distance was $220$ miles?
1
vote
0answers
41 views

Total Ordering on the Rationals

I am trying to prove that $\leq$ is a total ordering on the set of all rationals. So I started with the definition of the $\leq$ in the rationals, which is that if $x$ and $y$ are in the rationals, ...
0
votes
1answer
36 views

Rotations of complex graphs

Let $c_1 = -i$ and $c_2 = 3$. Let $z_0$ be an arbitrary complex number. We rotate $z_0$ around $c_1$ by $\pi/4$ counter-clockwise to get $z_1$. We then rotate $z_1$ around $c_2$ by $\pi/4$ ...
0
votes
0answers
11 views

Inverse and asymptotes

Given $f(x) = (2x+3)/(3x+2)$: A) Find the formula for $f^{-1}(x)$. B) Find the vertical and horizontal asymptotes of $f^{-1}(x)$. I am having trouble solving for the inverse, because i get an xy ...
2
votes
1answer
43 views

Rotation in the complex plane

The function $f(z) = \frac{(-1 + i \sqrt{3}) z + (-2 \sqrt{3} - 18i)}{2}$ represents a rotation around some complex number $c$. Find $c$. Hello, I am having some trouble trying to do this problem. ...
0
votes
3answers
86 views

Why is $3^{n+1} = 3^n + 2\cdot 3^n$?

I don't see how this equation holds true $3^{n+1} = 3^n + 2\cdot 3^n$ especially when there is a $2\cdot 3^n$ on the right equation.
0
votes
0answers
7 views

Expressing weighted fraction as sum of weighted fractions

Suppose we are given $z = W \cdot \frac{\sum_i x_i}{\sum_i y_i}$ Is it possible to express $z$ in the following form?: $z \overset{?}{=} w \cdot \sum_{i}\frac{x_i}{y_i}$ How could we find $w$ in ...
1
vote
0answers
34 views

Is this chain of inequalities correct?

Is this chain of inequalities correct? If not how to make it works? $$\frac{\ln \left( 1+x^3+y^3 \right)}{\sqrt{x^2+y^2}} \le \frac{\left( x^3+y^3 \right)}{\sqrt{x^2+y^2}} \le \frac{ \left( ...
-1
votes
3answers
72 views

What is the easiest way to rearrange a formula to change the subject? [closed]

I really fumbled at arranging formulas, reading and following lots of resources I got no nowhere. For simple formulas it's pretty much easy, but for $y = \frac{1}{1-x^2}$ this is really tough. I ...
0
votes
1answer
33 views

Closed under vector addition and scalar multiplication

If $S=\{[y\cdot z,y,z]^T \mid y, z \in \mathbb{R}\},$ is $S$ closed under addition and scalar multiplication? This one confuses me as there are not restrictions to what part of $\mathbb{R}^3$ our ...
0
votes
5answers
49 views

Cubic Equation. (Factorisation)

I'm given this question, factorise $4x^3-7x-3$. Is this answer acceptable? $(x+\frac{1}{2})(x-\frac{3}{2})(x+1)$.
-1
votes
0answers
30 views

Division-free differentiation of $\frac1x$

Given $a>0$, $~b=\frac1a$ and $c>0$ is it possible to calculate $\frac{1}{a+c}$ in such a way to avoid divisions? The solution can be approximated, but the percent error must be less than 1% ...
0
votes
4answers
42 views

Eliminating $t$ in the solution of a Differential Equation

My task is to show that the trajectories of the system: $\frac{dx}{dt}=y$, $\frac{dy}{dt}=x$ are hyperbolas given by $H(x,y)=y^2-x^2=c.$ Solving the above system I got: $x=c_1e^t+c_2e^{-t}$, ...
7
votes
10answers
1k views

Caden has 4/3 kg of sand which fills 2/3 ​​ of his bucket. How many buckets will 1kg sand fill?

I have already finished Calculus II but I go back and practice the basics on Khan Academy. This problem confuses me conceptually every time. I know what the answer is, but I am having a hard time ...
18
votes
7answers
2k views

Why aren't these negative numbers solutions for radical equations?

I was working on radical equations and I came across a few problems where I got answers that worked when I checked, but were not listed as solutions. My teacher's only explanation was, "just because." ...
0
votes
0answers
38 views

Basic algebraic simplification

How is this equation: $$\frac{1}{y+z} + \frac{z-x}{(y+z)^2} + \frac{-x-y}{(y+z)^2}$$ simplified to $\frac{2(z-x)}{(y+z)^2}$?
1
vote
1answer
65 views

Simplifying Sum

How would one show that $$ \sum_{i=0}^n\binom{n}{i}(-1)^i\frac{1}{m+i+1}=\frac{n!m!}{(n+m+1)!} ? $$ Any hint would be appreciated. Note: I tried to recognize some known formula, but since I don't ...
11
votes
5answers
2k views

How would multiplying money work?

This is a very silly question since nobody will actually do this because it makes very little sense in the real world but I just want to know how would it actually work if possible. For example let ...
0
votes
2answers
44 views

2xy-y=4+5x, what rule is being applied in the next step, that leads to (2x-1)y=4+5x

So the question says it all, I am sure that's the next step, and I also confirmed it with Wolfram Alpha, I am trying to calculate the inverse of a function, but I have a memory loss of what rule is ...
0
votes
1answer
32 views

Finding a basis for a subspace. Do i always need to test linearly independence?

Where the subspace is contained in {[5r-3s;2r;0;-4s] is an element of R^4: r and s are scalars} The generating set that can make up all of the input is {[5;2;0;0], [-3;0;0;-4]} This is only a ...
2
votes
2answers
68 views

Volume of a parallelepiped, given 8 vertices

Given the eight vertices $(0,0,0)$, $(3,0,0)$, $(0,5,1)$, $(3,5,1)$, $(2,0,5)$, $(5,0,5)$, $(2,5,6)$, and $(5,5,6)$, find the volume of the parallelepiped. I'm having trouble finding the 1 vertex ...
0
votes
3answers
54 views

Setting two equations equal to each other

I am trying to decipher the steps needed to determine the intersection point between curves represented by the two following equations: $$y= 2\;x+2$$ $$y= \dfrac{128}{(x+1)^2}$$ I know that they ...
2
votes
1answer
73 views

Proving a certain function is injective

I have found the following exercise on an exam for Geometry three dating to a past year. Let $F(u,v)=((2-v\sin\frac{u}{2})\sin u,(2-v\sin\frac{u}{2})\cos u,v\cos\frac{u}{2})$, with ...
0
votes
3answers
65 views

Confusing algebraic solution…

Hi everyone I'm a bit confused how they got that final value of M. Any ideas?
0
votes
1answer
34 views

An elephant and a lion are currently 1 mile apart. The elephant runs directly away from the lio

An elephant and a lion are currently $1$ mile apart. The elephant runs directly away from the lion at $19$ miles per hour, while the lion runs directly towards the elephant at $24$ miles per hour. How ...
0
votes
2answers
14 views

Expansion of Brackets

Case 1: $(a.b)(c.d)$ Case 2: $(e+f)(g+h)$ In both cases if you a value for the for each letter just calculate each bracket and multiply by the value of the second bracket. If no values are given ...
0
votes
1answer
26 views

Maximization of a function in an interval

I am writing a computer program where I have $x$ real positive varying in the domain $[\sqrt{U}, U]$. I want the value of $x$ which maximizes: $$ (1+ \sqrt{U}) - \frac{\sqrt{U}-1}{U-\sqrt{U}} x - ...
-3
votes
1answer
105 views

Why a line is said to have infinite number of points? [duplicate]

Why a line is said to have infinite number of points? Is this so because a line is ever lasting or we can not count how many points does it have? Finite means: Having an end. Infinite means: No end! ...
4
votes
1answer
59 views

Time and distance: Police and a thief with a twist.

A thief was given a head-start of 15 hour. The velocity of the thief being 4 km/hr and the police chasing after him be 5 Km/hr. A dog is moving to and fro between the police and the thief, starting ...
2
votes
5answers
93 views

How to sum $\sum_{k=1}^n (k+1)(k)(k-1)$

Is there an intelligent way to do this sum without using sums of cubes and sums of squares? $$\sum_{k=1}^n (k+1)(k)(k-1)$$
1
vote
2answers
29 views

Factorizing expressions

I am having trouble solving this problem $81f^2- \dfrac{9}{e^2}$. How do you begin when solving this problem? Do you move $f^2$ by replacing the $9$ and vice versa and does the minus change to plus?
1
vote
2answers
70 views

Derivation for the general cubic formula

It's a long equation, and Wikipedia writes it to be $$x_k = -\frac{1}{3a}(b + u_kC + \frac{\Delta_0}{u_kC}), \quad k \in \{1,2,3\}$$ But there is no derivation of it. The sources I've read so far ...
3
votes
2answers
157 views

Simple maths and a typo: impossible answer?

Long story short: While doing some simple math exercises, I came across one that seemed impossible. Days later I decided to search the web for it, and found out there was a typo, putting the $^2$ ...
2
votes
2answers
26 views

Strange formula for radar horizon, does anyone know where it comes from

I am in Electronics Technician "A" School in the Navy and we are learning the basics of radars. In one module, we were exposed to this formula which confuses me. If someone has an inkling as to why it ...
0
votes
1answer
33 views

Given $\tan x +\cot x = 3$ and $x$ is in first quadrant. Find $\sin x$.

Simplifying, I have $$\frac{1}{\sin x\cdot \cos x} = 3$$ I have tried many manipulations but did not get the answer. Point me the right direction to the solution. (This problem is in the beginning ...
0
votes
0answers
21 views

Parallelogram with vertices 0, Xa, Xb, Xa+Xb (X is matrix, a and b are vectors)

There is a paralellogram with vertices 0, a, b, and a+b, whose area is $34$. What is the area of the parallelogram which has vertices 0, Xa, Xb, and Xa+ Xb, where X = \begin{pmatrix} 3 & -5 \\ -1 ...
-2
votes
1answer
16 views

Find the coordinates of the points at which the line tangent to the graph of the function f(x) has the designated slope.

Find the coordinates of the point(s) at which the line tangent to the graph of the function f(x) has the designated slope. How do you find the coordinates if the derivative ends up having an $x^3$ ...
0
votes
2answers
36 views

Trisecting a line in the complex plane

We have $x = 11-13i$ and $y = 35-i$. $a$ is a complex number which trisects the line segment joining $x$ and $y$. $a$ is also closer to $x$ than $y$. Find $a$. I'm not sure where to start. Would a ...