Questions tagged [algebra-precalculus]

For questions about algebra and precalculus topics, which include linear, exponential, logarithmic, polynomial, rational, and trigonometric functions; conic sections, binomial, surds, graphs and transformations of graphs, solving equations and systems of equations; and other symbolic manipulation topics.

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Is it true that the curves $f(x,y)=0$ and $f(y,x)=0$ can only intersect on the line $x=y$?

Assume that the function $f(x,y)$ is not symmetric in $x,y$. Consider the curve $f(x,y)=0$ in $\mathbb{R}^2$. Is it always true that $f(a,b)=f(b,a)=0$ only if $a=b$? It does appear to be true when I ...
Ryan Hendricks's user avatar
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55 views

What are all possible set of $x,y,z$ such that $|x|=|y|=|z|=1$ and $x^3+y^3+z^3=-xyz$?

In this question, I asked for all the possible values of $|x+y+z|$ such that $x,y,z$ are complex numbers, $|x|=|y|=|z|=1$, and $x^3+y^3+z^3=-xyz$, and the answer was only $1$ and $2$. I found these ...
pie's user avatar
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1 answer
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Function Investigation proof verification

https://pasteboard.co/WMejlmetcznl.png This is the question We have, \begin{gather} f( m+n) \ =\ f( m) \ +\ f( n) \ +\ mn( m\ +\ n) \notag\\ \end{gather} And we have to prove, \begin{gather*} f( n) \ =...
Aritro Shome's user avatar
4 votes
3 answers
71 views

Roots of quartic equation - given product of two roots, find missing coefficient

The quartic equation $ax^4 + bx^3 + cx^2 + dx + e = 0$ has roots $\alpha, \beta, \gamma, \delta$. Given that $\alpha \beta = p$ find the value of $k$ So I have deduced that $\gamma \delta = \frac{e}{...
PhysicsMathsLove's user avatar
1 vote
2 answers
53 views

Solving $2mx^{2m+1}-(2m+1)x^{2m}+1=0$

I set up a problem for myself and came to this equation and I'm wondering if it can be solved algebraically. The variable $m$ is a whole number and we're solving for $x$. $$2mx^{2m+1}-(2m+1)x^{2m}+1=0$...
MumboJumbo's user avatar
1 vote
0 answers
13 views

Let $x,y\in\mathbb{R}^n$, $p\geq1$, determine a bound for $|x+y|^{p-1}$.

Let $x,y\in\mathbb{R}^n$, $p\geq1$, determine a bound for $|x+y|^{p-1}$. My approach: I know that if $x,y\geq0$ and $q\geq1$ then $(x+y)^{q}\leq 2^q(x^q+y^q)$, but in the case when $q=p-1$ I dont know ...
Darek_'s user avatar
  • 41
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0 answers
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How to calculate this expression (-1)^(2/10)? [duplicate]

What is (-1)^(2/10)? If we do ((-1)^2)^(1/10) than it is 1 but if we do (-1)^(1/5) then it is -1. So how to solve it?
Sean Nguyen's user avatar
1 vote
1 answer
45 views

Coprime cycle problem help (precalculus-level) [closed]

Problem: a set of integers has a coprime cycle if it can be listed in a cyclic list of the form $(a_1, a_2, a_3, \ldots, a_n)$, where each element of the set appears exactly once and $\gcd(a_i, a_{i+...
Mintylemon66's user avatar
0 votes
1 answer
32 views

Problem with the use of monomials: determine a formula

Andrea has $x$ euros, Marco has $y$ euros more than Andrea but $z$ euros less than Luke. Determine the formula that expresses the total sum $s$ owned by the three friends. Since Andrea has $x$ euros, ...
Sebastiano's user avatar
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1 vote
1 answer
59 views

What is the range of the function $\frac{\sqrt{x^2 + 4x + 3}}{x-5}$

I want to find the domain and range of the function $y = \frac{\sqrt{x^2 + 4x + 3}}{x-5}$. I found the domain to be $(-\infty,-3] \cup [-1, 5) \cup (5, \infty)$. I also found the inverse of the ...
zlaaemi's user avatar
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3 votes
2 answers
101 views

Calculate the range of the function $f(x)=-x^2+6x$ with the domain $[2,5]$

Consider the function $f(x)=-x^2+6x$ with the domain $[2,5]$. Now calculate the range of the function according to its domain. Here is my solution: The domain is $2\leq x\leq5$ so I can wrote: $2\...
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0 votes
0 answers
50 views

A functional inequality over integers

Given positive integers $N$ and $n$, we define a function $f(x)$ as follows: $$f(x):=\frac{x^2-3x}{2}~\text{ when } x\leq N,$$ and $$f(x):=\frac{1}{2n}x^2+\frac{n-4}{2}x~\text{ when } x>N.$$ ...
Kim's user avatar
  • 201
2 votes
1 answer
65 views

General polynomial equation I've made

For quite a while, from time to time I worked on a way to create a general nth degree/polynomial equation and I think it'd be great to show it to others as I think it doesn't have any mistakes and it ...
ssvv's user avatar
  • 23
0 votes
0 answers
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Solving four simultaneous equations

I am thinking how to solve the four simultaneous equations for $x,y,z,u$ where $\beta, c, a, b$ are arbitrary real numbers. $$\frac{a-x}{b} - z - u = 0,\\ x - min(\beta u, c y) =0,\\ x -y = 0,\\ x-1 =...
BAYMAX's user avatar
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0 votes
1 answer
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Identify $\mathbb{Z}_n$ with $ [- \frac{n}{2},\frac{n}{2})$

I have read somewhere that we can identify the group $\mathbb{Z}_n= \lbrace 0,...,n-1\rbrace $ with the interval $[- \frac{n}{2},\frac{n}{2})$ but there was no justification for this, does anyone have ...
Asma's user avatar
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1 vote
1 answer
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The given answer is 1,800, whereas I found it to be 1,600. Please explain how.

In an engineering college of 10,000 students, 1,500 like neither their core branches nor other branches. The number of students who like their core branches is 1/4th of the number of students who like ...
Musicmaniac's user avatar
3 votes
4 answers
69 views

Proving $\sum_{i=0}^n (-1)^i\binom{n}{i}\binom{m+i}{m}=(-1)^n\binom{m}{m-n}$

I am trying to prove the following binomial identity: $$\sum_{i=0}^n (-1)^i\binom{n}{i}\binom{m+i}{m}=(-1)^n\binom{m}{m-n}$$ My idea was to use the identity $$\binom{m}{m-n}=\binom{m}{n}=\sum_{i=0}^n(-...
Hjlmath's user avatar
  • 95
0 votes
2 answers
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Where Did I make a mistake in my deviation for the Period of a Pendulum Formula

Imagine a Pendulum with String Length L, and Angle θ, which forms arclength S (Will be used later in the derivation.) First V = $\sqrt{2gh} $ (Height is measured straight down) KNOWN TO BE A ...
Kyotiq's user avatar
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0 votes
1 answer
58 views

How does $ \frac{\sqrt{x}\left(2ax+b\right)-\frac{ax^2+bx+c}{2\sqrt{x}}}{x} $ become $ \frac{3ax^2+bx-c}{2x^{3/2}} $?

The original question was: Differentiate the given function with respect to x: $$ f(x) = \dfrac{ax^2+bx+c}{\sqrt{x}} $$ I simplified till $$ \dfrac{\sqrt{x}\left(2ax+b\right)-\frac{ax^2+bx+c}{2\sqrt{x}...
Thomas's user avatar
  • 9
-2 votes
0 answers
54 views

The sum of five natural numbers, two of which are greater than 40, is 127. Accordingly, what is the largest of these numbers?

What I think the question wanted to convey is that we should find the largest possible value in $a,b,c,d,e$ so that their sum is $127$, but so that two of them are greater than $40$. The given (...
THE_CRANIUM's user avatar
0 votes
3 answers
54 views

Difference of two greatest integer functions

If $[x]$ is the greatest integer less than or equal to x, then from the graph of the greatest function, and splitting into a case where x is an integer, and another where it is not, it is "...
Starlight's user avatar
  • 1,548
0 votes
0 answers
58 views

Plot a function graph

$y=\frac{1-\operatorname{tg}^2 2 x}{1+\operatorname{tg}^2 2 x}$ Photo
Михаил Байраков's user avatar
-3 votes
0 answers
33 views

Express the integrand as a sum of partial function and evaluate the integral [closed]

The integral is - integral x+3/2x^3-8x dx Show all the work for better understanding
coder's user avatar
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0 answers
108 views

An analytic solution to solve $x^9=3^x$

I want to find a way to solve $x^9=3^x$ analytically, for two roots. one of them can be found below $$x^9=3^x\\(x^9)^{\dfrac {1}{9x}}=(3^x)^{\dfrac {1}{9x}}\\x^ { \ \frac 1x}=3^{ \ \frac 19}\\x^ { \ \...
Khosrotash's user avatar
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-5 votes
0 answers
58 views

Let $n$ be an integer. Show that if $n$ has a multiplicative inverse in $\mathbb{Z}$, then $n=1$ or $n=-1$. [closed]

Let $n$ be an integer. Show that if $n$ has a multiplicative inverse in $\mathbb{Z}$, then $n=1$ or $n=-1$. This seems very simple, but I could not come up with a proof that would satisfy me.
Sarthak Hazra's user avatar
-4 votes
0 answers
40 views

Prove the trigonometry identity [closed]

$\frac{\left(\cos (2 \pi-\alpha)+\sin \left(\frac{3 \pi}{2}+5 \alpha\right)\right)\left(\cos \left(\frac{\pi}{2}-\alpha\right)-\sin (\pi+5 \alpha)\right)}{1+\sin \left(\frac{3 \pi}{2}-6 \alpha\right)}=...
Михаил Байраков's user avatar
0 votes
1 answer
116 views

How analytically solve $ax^2-b \sqrt{x}+c=0$

How analytically solve $ax^2-b \sqrt{x}+c=0$, where a, b, c are positive constants? mathematica gives long complex solutions but not sure if it's reliable.
feynman's user avatar
  • 147
2 votes
2 answers
48 views

Steps WA used to simplify $\left( \frac{\cos \beta \sin \beta \cos\phi +\sin \beta}{\sin^2 \phi + \sin^2\beta \cos^2 \phi} \right)^2\cos \phi$

I have the following expression: $$\left( \frac{\cos \beta \sin \beta \cos\phi +\sin \beta}{\sin^2 \phi + \sin^2\beta \cos^2 \phi} \right)^2\cos \phi$$ Wolfram Alpha yields a simplified form of this ...
rdemo's user avatar
  • 129
0 votes
2 answers
65 views

Show that the sequence $(1+N^2)/[N(1+N)]-5/6$ is positive and increasing

I have the function \begin{equation*} f(N)= \frac{1+N^2}{N(1+N)}-\frac{5}{6}, \end{equation*} where $N\in\mathbb{Z}_{++}/\{1, 2\}$. I want to show that $f(N)>0$ and that $f(N+1)>f(N)$ for ...
Hal_Incandenza's user avatar
0 votes
0 answers
32 views

re-scaling function converges to another function?

Consider: $$\Psi(x)= \sum_{n=1}^\infty \exp\bigg(\frac{\log n}{\log x} \bigg)=\zeta\bigg(\frac{-1}{\log x}\bigg),~~~~~~~x\in(1/e,1) $$ for $\zeta$ being the Riemann zeta function, and a re-scaling ...
John Zimmerman's user avatar
0 votes
2 answers
30 views

Showing $\tan x = \frac{\sin\alpha\sin y}{1 - \sin\alpha\cos y}$, given $\sin x = \sin \alpha\sin (x + y)$

I've been struggling with this for some time but can't prove what question asks. Question is as follows: If $\sin x = \sin \alpha\sin (x + y)$, prove that $$\tan x = \frac{\sin\alpha\sin y}{1 - \sin\...
GR L's user avatar
  • 319
-3 votes
2 answers
85 views

Solving $(x^3 -2x-4)^{1/3}+(x^2 -x -1)^{1/2}=1$ [closed]

Can anybody help me to solve this problem,even I can't try with it! I want all solutions for this problem: Find $x$ in $$(x^3 -2x-4)^{1/3}+(x^2 -x -1)^{1/2}=1$$
THE Thrones's user avatar
1 vote
2 answers
99 views

How many times is the light bulb turned on?

A light bulb is automatically turned off for 21 secs. Next, it lights up automatically for 15 secs. Once again, it is automatically turned off for the following 21 secs. The process keeps repeating. ...
x89's user avatar
  • 189
3 votes
2 answers
242 views
+50

algebraic equation with powers

The original statement of the problem : Solve in $\mathbb R$ the following equation : $$ a^{log_b x^2 } + a^{log_x b^2 } = a^{1+log_b x } + b^{1+log_x b } $$ where $a,b>0$ and $b \neq 1$ A more ...
Last X's user avatar
  • 223
0 votes
2 answers
71 views

Why are there two different ways to solve the same problem but two different answers?

I was talking to a professor, and we both were stumbled on the problem of: $\sqrt{\sqrt{x^{2}}-1}=2$ My professor's thought process was that you would immediately square both sides to get $\sqrt{x^{2}-...
Payden 's user avatar
1 vote
3 answers
66 views

Attempt at creating a formula relating debt, payments and interest

I tried writing down a formula relating a given debt and interest to the periodic payments and number of payments. So let's say someone starts off with a debt of $D$. The periodic interest is $r$ (for ...
HappyDay's user avatar
  • 1,027
2 votes
2 answers
94 views

Is there a spelling mistake or am I missing something

Here, $[ \cdot]$ is $\lfloor \cdot \rfloor $ floor function. N $ \in N$. Where did $\frac{[Nx]} N + \frac{1}{2N}$ came from and how does $x$ differs by $\frac{1}{2N}$. Shouldn't it be $\frac{1}N$ if ...
Yugant Shewale's user avatar
5 votes
2 answers
136 views

Show $\root{-e}\of{e}<\ln2$ without a calculator

I tried manipulating the expression to come up with inequalities such as $1<e^{\frac{1}{e}}\ln2$. One idea I have is to show that $\lim_{x\to\infty}\left(\frac{1}{x}+\ln2\right)\left(\frac{1}{x}+1\...
Dylan Levine's user avatar
  • 1,575
0 votes
1 answer
109 views

Alternative method for solving $|3x-9|+|7x-8|-||20x-13|-3x|-3=0$

Recently I came across the following equation: $$|3x-9|+|7x-8|-||20x-13|-3x|-3=0.$$ An obvious approach would be to divide it into cases and solve for each interval for each mod involved. Another ...
orionbuff_62's user avatar
2 votes
1 answer
82 views

show that an expression $E(x,y)$ does not depend on x

the question Let $a,b,d,e$ be real numbers with $a>d$ and $c=a^2+b^2, f=d^2+e^2, m=\frac{b-e}{a-d} , n=\frac{bd-ae}{a-d}$. Show that if $x\in [-a,-d] $ and $y=mx+n$, then $$E(x,y)=\sqrt{x^2+y^2+2ax+...
IONELA BUCIU's user avatar
  • 1,179
-5 votes
0 answers
46 views

how to solve f(x)*f(f(x)+1/x)=1 if f(x) is increasing ft? [closed]

Problem give me condition that f(x) domain x>0, and f is increasing ft only. No given Differentiability and Conditions for the existence of an inverse function. I thought ft is t/x, so if my guess ...
growww's user avatar
  • 1
-4 votes
1 answer
51 views

Find $a$ and $b$ if $\lim _{x\to 3}\frac{3x^2+ax+b}{x^2-2x-3} = \frac{5}{2}$ exists [closed]

How to find $a$ and $b$ of this expression or limit $$\lim _{x\to 3}\frac{3x^2+ax+b}{x^2-2x-3}=\frac{5}{2}$$ I don't really have any clue on how to solve this
Ceerwin Yamson's user avatar
0 votes
0 answers
23 views

Prove special case of two class softmax is equivalent to logistic function

A textbook I'm reading asks to prove that the special case of a two value softmax is equivalent to a sigmoid/logistic function. It defines the softmax as: $$ \frac{e^{H_t(a)}}{\sum_{b=1}^ke^{H_t(b)}} ...
foreverska's user avatar
1 vote
1 answer
85 views

Simplify $\log((10\cdot 8 )^{\frac{1}{2}} \times (0.24)^{\frac{5}{3}} \div (90)^{-2})$

Simplify $\log((10\cdot 8 )^{\frac{1}{2}} \times (0.24)^{\frac{5}{3}} \div (90)^{-2})$ $\Rightarrow \log(10\cdot 8)^{\frac{1}{2}}+\log(0.24)^{\frac{5}{3}}-\log(90)^{-2} \tag{1}$ $\Rightarrow \dfrac{1}...
ronald christenkkson's user avatar
0 votes
6 answers
130 views

Why is $2^n +1=(2+1)(2^{n-1} - 2^{n-2} +2^{n-3}-\ldots+1)$ for odd $n$ [closed]

This transformation is only a part of solution of my problem but the most significant one. I need to show that $2^n+1$ is divisible by $3$ for odd $n$’s. I have exercised polynomial transformation for ...
GreyCow's user avatar
  • 25
0 votes
1 answer
31 views

Need hints (advice) to prove $(\forall a,b,c,d \in \mathbb{R}) (a < b \wedge c<d) \Rightarrow ad+bc < ac +bd$

I'm trying to prove this ( source : my uni's textbook says that it's trivial). $$(\forall a,b,c,d \in \mathbb{R}) (a < b \wedge c<d) \Rightarrow ad+bc < ac +bd$$ So far, I've managed to get ...
runtotherescue's user avatar
0 votes
0 answers
85 views

Closed form for a finite sum

Define $$f_n(x_1,x_2,x_3,x_4)=\sum_{s_1=0}^n a_{n,s_1,s_1,s_1,s_1} x_1^{s_1}x_2^{s_1}x_3^{s_1}x_4^{s_1}+\sum_{s_i\neq s_j, \forall i\neq j,s_i\in[0,n]} a_{n,s_1,s_2,s_3,s_4} x_1^{s_1}x_2^{s_2}x_3^{s_3}...
Max's user avatar
  • 266
2 votes
0 answers
124 views

I need help understanding the intuition behind a solution. [closed]

Suppose that $$x + y + z = 203.$$ What would be the maximum value of $$x + (xy)^{1/2} + (xyz)^{1/3}?$$ The solution is to assume that $$\frac43\cdot (x+y+z)\ge x+(xy)^{1/2}+(xyz)^{1/3}$$ and prove it ...
ConfusedConfucius's user avatar
2 votes
1 answer
66 views

Finding the perimeter of an equilateral triangle given height

Given an equilateral triangle with vertices $A,B,C$, find the perimeter of the triangle if the height of this triangle is $8\sqrt{6}$. I do not have a visual image for this problem, because I cannot ...
Microtask's user avatar
  • 321
-2 votes
0 answers
67 views

Prove that $\frac{\alpha}{\beta}\cdot\frac{\gamma}{\delta}=\frac{\alpha \gamma}{\beta \delta}$ [closed]

How in the world am I supposed to probr this obvious thing? $$\left(\forall \alpha, \beta, \gamma, \delta \in \mathbb{R}, \beta \neq 0, \delta \neq 0\right) \frac{\alpha}{\beta}\cdot\frac{\gamma}{\...
runtotherescue's user avatar

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