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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9 views

How do I factor z^3+2z^2-3z+20?

I just can't factor this equation: z^3+2z^2-3z+20 How do you do it? Oh my god!
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4answers
44 views

How do I factor $z^4+2z^3+4z^2+2z+3$?

It seems so that I need help factoring this equation $z^4+2z^3+4z^2+2z+3$. Both complex and real
0
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4answers
27 views

Intuitive way of solving this inequality: $x^2>y^2 \wedge x>0 \Rightarrow x>y$

I have to prove that $x^2>y^2 \wedge x>0 \Rightarrow x>y$. I decided to do this: $x^2>y^2 \Leftrightarrow \sqrt{x^2}>\sqrt{y^2}\Leftrightarrow |x|>|y| \Leftrightarrow x>|y| \vee ...
1
vote
1answer
13 views

net profit percentage

A shopkeeper mixes 25% kerosene to his petrol and then he sells the whole mixture at the price of petrol. If the cost price of kerosene be 50% of the cost price of petrol what is the net profit ...
0
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3answers
26 views

Solutions of a symmetrical system of $3$ equations in $3$ unknowns

I've tried to solve the following system, but the process seems a bit awkward to me. Any hints? $x+y+z=6$ $xyz=6$ $xy+yz+zx=11$
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1answer
41 views

property of two numbers such that their exponential and product are the same

Here is a equation: $$ a^b = ab$$ [ that is, $a$ raised to $b$ is equal to $a$ times $b$ ] Find all real values of $a$ and $b$ where $a$ is not equal to zero, $b$ is not equal to one. ...
2
votes
3answers
27 views

Finding constants with a piecewise continuous function

Let $$f(x) = \begin{cases}\frac{3x^{2}+ax+a+3}{x^{2}+x-2}, & x \neq -2 \\ b & x= -2 \end{cases}$$ Given that $f(x)$ is continuous at $x=-2$, find $a$ and $b$. Above is the problem I'm ...
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1answer
46 views

Integral with quadratic square root inside trigonometric functions

Is there anyway to solve $\displaystyle \int t \frac{\sin \left(\frac{t}{2} \sqrt{ a \left(t+ \frac{b}{2a}\right)^2-\frac{b^2-4ac}{4a}}\right) }{ \sqrt{ a \left(t+ ...
2
votes
2answers
54 views

Help with finding the real zeros of a polynomial

$$P(x)=x^4-6x^3+4x^2+15x+4$$ Steps I took: Possible zeros are: $$(x+1)(x-1)(x+2)(x-2)(x+4)(x-4)$$ Used synthetic division to find which zero is an actual zero: (I apologize for the graphical ...
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1answer
30 views

Show that $\frac{a^{n+2} - 1}{a - 1}=\frac{a^{n+1} - 1}{a - 1} + (n + 1)$ [on hold]

How can I show that \[ \frac{a^{n+2} - 1}{a - 1}=\frac{a^{n+1} - 1}{a - 1} + (n + 1) \]
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3answers
33 views

Proving that $\dfrac{\tan(x+y)-\tan x}{1+\tan(x+y)\tan x}=\tan y$

Edit: got it, silly mistakes :) I need to prove that $\dfrac{\tan(x+y)-\tan x}{1+\tan(x+y)\tan x}=\tan y$ $$=\frac{\tan x+\tan y-\tan x+\tan^2x\tan y}{1-\tan x\tan y+\tan^2x+\tan x\tan y}$$ ...
-2
votes
1answer
63 views

how can i prove $\sum |a_i||b_i|\le(\sum |a_i|)|b|$? [on hold]

Why $\sum |a_i||b_i|\le(\sum |a_i|)b$ ? Where $|a_i|$ and $|b_i|$ are real numbers and b=$\sqrt{\sum(b_i)^2}$ How do I prove the above inequality? Or what is the theorem that proves that?
4
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4answers
132 views

Determine whether $\sqrt{\sqrt{5}+3}+\sqrt{\sqrt{5}-2}$ is rational

I need some help with this problem: the task is to determine if the number $\sqrt{\sqrt{5}+3}+\sqrt{\sqrt{5}-2}$ is rational or not. Unfortunately I barely have an idea how to start and hence would ...
0
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2answers
59 views

How do I factor $x^6+x^3+4$? Both real and complex

I can't seem to figure out how to factor this: $x^6+x^3+4$ Supposed to give both the real and complex factors
0
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2answers
57 views

Show that $f:\mathbb{R}-\{2\}\to\mathbb{R}-\{5\}$ with $f(x)=\frac{5x-1}{x-2}$ is bijective

Can anyone please help to explain the question and what actually $f: \mathbb{R} - \{2\}$ means ?? I know that bijection means one to one function and onto both. Any idea to start up with this ...
1
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1answer
44 views

Trying to find an $\arctan(x/y)$ identity.

I have this equation : $$\theta = \arctan\left(\tfrac xd\right) + \arctan\left(\tfrac yd\right).$$ $\theta$ is an angle and I am trying to express $d$ as a function of $\theta$. So is there a way ...
-3
votes
1answer
36 views

When equality holds in an inequality

I am working on a class project, the passage I quoted in here is from a book Complex Numbers & Geometry by Hahn. For any four complex numbers $a$, $b$, $c$, $d$, the following identity is easy ...
2
votes
0answers
65 views

Prove $\frac {1}{(a-b)^{2}} + \frac {1}{(b-c)^{2}} + \frac {1}{(c-a)^{2}}=1$ [duplicate]

Given that $a,b,c$ are distinct real numbers such that $a^{2}*(1-b+c)+b^{2}*(1-c-a)+c^{2}*(1-a+b)=ab+bc+ca$ First, I tried to expand the bottom of what we need to prove. ...
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0answers
59 views

Prove that $\sqrt{n}$ is irrational [on hold]

Question: Using fundamental theorem of integers and the fact that every natural number that is not prime, prove that $\sqrt{n}$ is irrational unless $n=m^2$ for some $m\in\mathbb N$. Here is how I ...
0
votes
1answer
17 views

Finding the vertical shift of a sinusoidal function

I'm currently studying sinusoids, I've been given a graph with a few key points and have been told to find a cosine function which fits it. When it comes to finding the vertical shift of the graph the ...
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1answer
25 views

Finding a nontrivial solutions in natural numbers.

Consider the equation for natural numbers $i,j,k,l:$ $$ (j^2-i^2) (k\cdot l)^2=2\, (l^2-k^2) (i\cdot j)^2. $$ I am trying to prove that it has no solution. To undertand why, let us first consider ...
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2answers
41 views

$m\cos^2{\theta} + n\sin^2{\theta} < l \implies \sqrt{m}\cos^2{\theta} + \sqrt{n}\sin^2{\theta} < \sqrt{l} $

Prove that $m\cos^2{\theta} + n\sin^2{\theta} < l \implies \sqrt{m}\cos^2{\theta} + \sqrt{n}\sin^2{\theta} < \sqrt{l} $ for every $m, n, l >0$.
2
votes
1answer
30 views

How to prove this assertion about $\mathbb{R}^k$?

Suppose $k \geq 3$, $x$, $y \in \mathbb{R}^k$, $|x-y| = d > 0$, and $r > 0$. Then how to prove the following assertions? (a) If $2r > d$, then there are infinitely many $z \in \mathbb{R}^k$ ...
0
votes
2answers
17 views

Function with exponent imaginary power

If we have $u=\frac{4c(e^{-is}-e^{is})}{(e^{-is}+e^{is})^2} \tag 1$ where c is a constant and s is a variable. Can we write $e^{is}$ in terms of u ? Means Can we write $e^{is}$ as $\psi(u)$ , a ...
0
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1answer
25 views

Calculate distance base on value that is vary

I am new to Mathematica. I am stuck at getting one specific value after calculating with value that is changing from 0.5 to 1.0 I have value 400 and from that I want to go to destination 720 using ...
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votes
2answers
42 views

Clarification of Identification [on hold]

This is more of an observation question. When you see $x$, In $f(x) = x^2$ And when you see $g(x) = x^3$ You automatically identify $x = x$ Wouldn't the $x$'s be off by a little bit? But ...
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0answers
36 views

Class 10 maths questions [on hold]

Solve the following pair of linear Question graphically :- 6x - y + 4 = 0 2x -5y = 8 (Shade the region bounded by the lines X and Y - axis) Prove that the ratio of the areas of two ...
0
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0answers
37 views

Solving a sets problem with fractions.

In a group of students, each student is taking a mathematics course or a computer science course or both. One-fifth taking a mathematics course are also taking a computer science course, and one-eight ...
0
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1answer
16 views

Rearrange equation with integrating factor

I'm trying to do the following in the middle of a huge question involving a differential equation - I need to rearrange this equation for t, but have no idea where to start. First image is the ...
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0answers
10 views

Help find the total distance [on hold]

The US presidential convoy stretches 4miles and is moving in to Boston. A special protective vehicle at the back was given a letter to deliver at the front of the convoy. After delivering the letter ...
0
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1answer
21 views

Combining two variables to form one variable

I have a variable $x$ which can be of value $0$ or $1$. Secondly, I have a variable $p$ which can go from $0$ to $1$. Secondly, if $x = 0$, then $p$ is also $0$. I want to make another variable which ...
0
votes
1answer
17 views

Derivates and Limits in the Same Problem are an Issue.

I am working on the following problem:- Evaluate lim x→1 [( x^1/4 - 1 ) / ( x^1/3 - 1 )] by relating it to the derivatives of functions. Now this is quite a ...
0
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2answers
26 views

Simplify a function with a square root as the numerator

How would I go about simplifying this: $$\frac{\sqrt{x^4 + 3x^2}}{x}$$ thanks!
0
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1answer
17 views

How to solve a limits problem with perplexing conditions?

Let $f$ and $g$ be two functions satisfying $f(x) > g(x)$ for $0 < |x − a| < h$. Suppose $\lim_{x\to a} f(x) = L$, $\lim_{x\to a} g(x) = M$. Then it is not hard to show that $M ...
0
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1answer
41 views

Why do we say let x = soething as opposed to x = something?

In algebra word problems, we use let x = some variable as opposed to x = some variable. I've been told that we can't use just ...
1
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2answers
86 views

An inequality with a weird condition

$a,b,c$ are distinct real numbers that $$(a^2)(1-b+c)+(b^2)(1-c+a)+(c^2)(1-a+b)=ab+bc+ca.$$ Prove $$\frac{1}{(a-b)^2}+\frac{1}{(b-c)^2}+\frac{1}{(c-a)^2}=1.$$ I have tried two different approaches to ...
1
vote
3answers
248 views

How to show this equals 1 without “calculations”

We have $$ \sqrt[3]{2 +\sqrt{5}} + \sqrt[3]{2-\sqrt{5}} = 1 $$ Is there any way we can get this results through algebraic manipulations rather than just plugging it into a calculator? Of course, $(2 ...
0
votes
4answers
53 views

Function $f(x)=x^2+1$. Find a function g with $(f∘g)(x)=x+5$ ??

Function $f(x)=x^2+1$. Find a function $g$ with $(f∘g)(x)=x+5$. My answer was $g(x) = x^{0.5} + 2$ but that's incorrect because you would still have $4x^{0.5}$ remaining that wouldn't fit the ...
0
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1answer
12 views

Algebra question, Cardano's method

We have the following information: $$a^3 + b^3 = -q $$ $$a^3b^3 = -\dfrac{p^3}{27}$$ Apparently this yields the equation $z^2 + qz -\dfrac{p^3}{27}$ of which $a^3$ and $b^3$ are the roots. Can ...
1
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1answer
40 views

Is there a tight upper bound on $\sum_{i=1}^n \sum_{j=1}^m \min(a\cdot i,b \cdot j)$

Is there a tight upper bound on $\sum_{i=1}^n \sum_{j=1}^m \min(a \cdot i,b \cdot j)$ for any $a,b \in \mathbb{R}^+$ For example one upper bound would be \begin{align} \sum_{i=1}^n \sum_{j=1}^m ...
0
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1answer
29 views

From an expression raised in a power of 2 to an expression raised in the power or 10

Is there a simple/"easy" way to convert a big number from a power of $2$ to a power of $10$ equivalent. Example: I had $2^{127}\cdot 1.9999999$ which I did the multiplication got the result and from ...
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0answers
25 views

Defining Variables [duplicate]

My last post got put on hold, here's a new try. My question is, in pure mathematics, all variables (alike) have the exact same meaning. In pure mathematics, $x = x$ always, and $u=u$ My question ...
0
votes
1answer
15 views

Polynomials Multiplication in One Variable

Is there an efficient way/algorithm to extract coefficients in product of K polynomials in one variable? Eg: $ P_1(x) = p_{10} + p_{11}x + ... + p_{1n}x^n$ $P_2(x) = p_{20} + p_{21}x + ... + p_{2n}x^n ...
0
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1answer
29 views

Does this loop run in $\mathcal{O}(n^4)$ time?

A double loop is given: int sum = 0; for (int i = 0; i < N*N; i++) for (int j = i; j < N; j++) sum++; My analysis: The inner loop runs $n$ ...
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2answers
26 views

compostition of functions

$f(x)=12(8x+3)$ $h(x)=(3x+19)−5$ Solve when $x=8$ I really need step by step directions to solve $f(h(x))$. Also can someone solve $h(f(x))$. I really need help this is so confusing. I am able to ...
15
votes
4answers
3k views

In primary school I was showed this. Why does it work?

When I was in primary school a teacher showed us the following exercise in arithmetic. Take any 3 digit number between 201 and 998 provided that the hundreds digit is bigger than the ones digit and ...
0
votes
1answer
40 views

An identity that is always an integer

If $$\frac{a}{b} + \frac{b}{c} + \frac{c}{a} \in \mathbb{Z}$$ and $$\frac{b}{a} + \frac{c}{b} + \frac{a}{c} \in \mathbb{Z}.$$ For any natural $n$, is the following true? ...
3
votes
1answer
65 views

Why do we say that $\sqrt{-0} = -0$?

According to wikipedia's page on signed zeroes, we agree that $\sqrt{-0} = -0$. I would always have guessed that it would be $0i$, as $(0i)^2 = 0^2*i^2 = 0 * (-1) = -0$. I know that my own ...
0
votes
2answers
36 views

Is my method of computing the running time correct?

Okay, so this is the code for which I need to compute the running time: ...
3
votes
4answers
110 views

How do you read the symbol “$\in$”?

A variable in an equation may be replaced by any of the numbers in its domain. The resulting equation may be either true or false. Here is another way to show ...