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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Consider a quadratic equation $az^2+bz+c=0$, where $a, b, c$ are complex number. Then Find condition for one purely imaginary root.

Consider a quadratic equation $az^2+bz+c=0$, where $a, b, c$ are complex number. Then condition that above equation has one purely imaginary root (A)$(a\bar b+\bar ab)(b\bar c+ \bar b c )+(c \bar a -\...
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1 vote
1 answer
25 views

is tan(π/2 - x)=cot(x) because tan in the 2nd area is "negative" and also cot(-x) equals to -cot(x)?

trying to figure out if that negative/positive calculation is true. and so at the end there will be two negative and the cot(x) will be positive. is that so? thank you.
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0 votes
0 answers
15 views

A transcendental function?

Let $(u_n)_n=(p_n/q_n)_n$ $\left(\text{with $\mathrm{GCD}(p_n,q_n)=1$ and }\ln|u_n|=o(2^n)\right)$ a sequence of rational numbers that is not ultimately zero. I want to prove that the function $f(z)=\...
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0 votes
2 answers
25 views

Using quadratic equations to solve ratio questions

There are two parts to the question but I have answered the first part. The exchange rate in 1964 was $x$ dollars to £1. An American tourist remembered that in 1952 he needed $1.5$ dollars more for ...
1 vote
2 answers
51 views

Compare $~a,b,c~$ with $~1,2,3~$ - Revisited. The virtue of meta-cheating

This is a self-answer question that revisits a question that the original poster deleted, themself. The original question is Let $0 \le a \le b \le c$ such that $a+b+c \geq 6,ab+bc+ca=11,abc=6$. ...
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0 answers
44 views

I can't figure out how the average can be proven here to be greater than the variable shown

I have a solution to a math problem and I can't figure out the algebra for the life of me. How is $$\frac{i+j}{2} > k$$ ? My intution tells me that since the average of $\frac{3}{2}$ and $\frac{2}{...
4 votes
1 answer
47 views

Let $T$ be the following set of ordered triplets,$T=\{(a,b,c):a,b,c\in N\}$. Find the number of elements in $T$ such that $L.C.M(a,b,c)=72$.

Let $T$ be the following set of ordered triplets,$T=\{(a,b,c):a,b,c\in N\}$. Find the number of elements in $T$ such that $L.C.M(a,b,c)=72$. My Attempt Let $a=2^{x_1}3^{y_1}$,$b=2^{x_2}3^{y_2}$ and $c=...
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2 votes
1 answer
43 views

Can I infer the hypotenuse given only $a > b$?

Today, I came up with a problem. The problem is this: Let $a$, $b$ and $c$ be the sides of a right-angled triangle, such that $a > b$. The nature of $c$ is unknown. Can I infer the hypotenuse given ...
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8 votes
4 answers
135 views

Bounds on the maximum real root of a polynomial with coefficients $-1,0,1$

Suppose I have a polynomial that is given a form $$ f(x)=x^n - a_{n-1}x^{n-1} - \ldots - a_1x - 1 $$ where each $a_k$ can be either $0,1$. I've tried a bunch of examples and found that the maximum ...
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2 votes
2 answers
26 views

How do we derive the expression?

I am studying a book which gives the following expressions: $\alpha = \frac{\beta_1 d}{2} + \frac{\beta_2 d}{2}$ and $\gamma = \frac{\beta_1 d}{2n} - \frac{\beta_2 d}{2n}$. Then it needs to solve for $...
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-3 votes
0 answers
23 views

How to develop the algebraic expression? [closed]

Find the development of the algebraic expression: $\frac{(2+3)(2^2+3^2)...(2^{2048}+3^{2048})+2^{4096}}{3^{2048}}$
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22 views

Does the Distributive Property of Multiplication Over Addition apply to Absolutely Convergent Infinite Sums

Does the Distributive Property of Multiplication Over Addition apply to Absolutely Convergent Infinite Sums For example, is it true that $\sum\limits_{n=1}^\infty (k\cdot a_n) = k\left(\sum\limits_{n=...
2 votes
2 answers
81 views

Understanding a proof that, if $|a-1|+|b-1|=|a|+|b|=|a+1|+|b+1|$, then the minimum value of $|a-b|$, over distinct reals $a$ and $b$, is $2$.

I saw this epic question in Advanced Problems in Mathematics by Vikas Gupta: If $$|a-1|+|b-1|=|a|+|b|=|a+1|+|b+1|$$ then find the minimum value of $|a-b|$, where $a$ and $b$ are distinct real numbers....
0 votes
1 answer
37 views

Help recreating algebraic reasoning on the Euclidean metric.

I am trying to recreate and formalize with Lean4 this hint from "Euclidean plane and its relatives" by Petrunin: $$ \sqrt{(x_{1} + x_{2}) ^ {2} + (y_{1} + y_{2}) ^ {2}} \leq \sqrt{x_{1} ^ {2}...
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-2 votes
0 answers
33 views

Inverse of a Function With a Specific Domain [closed]

What is the Inverse of a Function $x^2 - 9$, For $x \geqslant -9$?
0 votes
1 answer
39 views

explanation of this cosine solution

I was studying Physics and I stumbled upon this equation: $$U_0 \cos{x}(\cos{x} - 1) = \alpha U_0$$ So you should get the value of x to do some other stuff. The equation doesn't seem to complex, ...
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3 votes
3 answers
324 views

Two Quadratics, the roots of one are the coefficients of the other and vice versa.... can they exist?

I spent hours on this... feel pretty pathetic. I read this on an old AMATYC exam: Let $\quad c, \quad$ and $\quad d\quad$ be the roots of $\quad x^2+ax+b$ and let $\quad a, \quad$ and $\quad b\quad$ ...
1 vote
2 answers
49 views

Finding the quotient and remainder when dividing $f = 2018X^{2019} - 2019X^{2018} + 1$ to $(X-1)^2$ [duplicate]

Let $f = 2018X^{2019} - 2019X^{2018} + 1, f \in \mathbb{R}[X]$. Find the quotient and the remainder when dividing $f$ to $(X-1)^2$. The answer to this problem is that $f = (X-1)^2(2018X^{2017} + ...
0 votes
0 answers
25 views

Hudde's cubic proof

I've been following the proof of Hudde's description of Cardano's method of cubic roots shown here https://proofwiki.org/wiki/Cardano's_Formula. Does anyone know where this proof comes from? I can't ...
0 votes
5 answers
96 views

Solve $x^2+4y^2+80 = 15x+30y \\ xy=6 $

Solve for $x$ and $y$: $$x^2+4y^2+80 = 15x+30y \\ xy=6 $$ I have no idea how to solve this. I tried setting $x=\frac{6}{y}$ and plugging in, but all I get is this: $(\frac{6}{y})^2+4y^2+80 = 15(\frac{...
-4 votes
2 answers
80 views

The velocity of light "c" a function of "e" ? Examine the following equations

The following 3 equations show that the velocity of light 'c' can be shown to have a connection to the mathematical constant 'e' Where c is the velocity of light in meters per second c =299,779,724 m/...
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0 votes
2 answers
49 views

Addition or subtraction of speeds?

Can we also take $(x+15)$ and $(x-15)$ in the following problem? Problem: The speed of a boat in still water is $15$ km/h. It goes $30$ km upstream and return downstream to the original point in $4$ ...
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0 votes
2 answers
60 views

When to use $\implies$ or $\iff$ [closed]

Whenever writing a solution for a equation when you go on to the next step what sign to use $ \implies $ or$ \iff$ like $\textbf{Scenario 1}$ $x^2=36 \implies x= \pm 6$ $\textbf{Scenario 2}$ $x^2 =36 \...
2 votes
2 answers
119 views

Solve simultaneous quadratic equations $ x^2-2xy=21$ and $xy+y^2 = 18$

Solve: $$ \left\{\begin{array}{l} x^2-2xy=21 \\ xy+y^2=18 \end{array} \right. $$ $\rightarrow$ $\left\{\begin{array}{l}x(x-2y)=21 \cdot \cdot \cdot(1) \\ y(x+y) = 18 \cdot \cdot \cdot(2)\end{...
-1 votes
0 answers
35 views

Find the Highest common factor of two polynomial [closed]

Find the HCF of: $$x^3 + x^3y, x^3 + y^3 $$ Answer is $$ x + y $$ but I don't know how. Can someone explain this?
4 votes
3 answers
91 views

Prove that the following inequality is true for all $m \in (0, 3)$

Prove that the following inequality is true for all $m \in (0, 3)$ $$\sqrt {{m^2} + 1} + \sqrt {{{\left( {3 - m} \right)}^2} + 1} \le \sqrt {\frac{{2\left( {4{m^2} - 12m - 15} \right)}}{{(m - 3)m}}} ...
-2 votes
0 answers
30 views

Underoot equation in a fraction [closed]

${\sqrt7\over \sqrt{16+6(\sqrt7)}-\sqrt{16-6(\sqrt7)}}$ [1]: https://i.stack.imgur.com/eO7fy.jpg
-1 votes
0 answers
36 views

How to combine sums? [closed]

TLDR (Question about the image) - Why is this correct? Where did "$-r$" go? Context - I had a proof (Cyclic Convolution Theorem) as part of my Signals & Systems course. As part of the ...
-3 votes
0 answers
32 views

Sum of Infinite recursion sequence [closed]

Let $a_1 = 2 $ , $a_{n+1} = a_n^2 - a_n +1 $ (n is a Natural number). Find $$\sum_{i=0}^\infty \frac{1}{a_n}$$
2 votes
1 answer
53 views

An unclear transition with logarithms

I watched a lecture, and it included the following segment. There are two functions: $f(n)$ and $g(n)$. Let's say we want to find any asymptotic relationship between those functions. We do a ...
0 votes
1 answer
66 views

Setting up and solving an equation of torque given some data

A heavy hatch on a ship is made of a uniform plate of steel that measures 1.2 m X 1.2 m and has a mass of 400 kg. The hatch is hinged along one side; it is horizontal when closed and opens upward. A ...
0 votes
1 answer
26 views

Prove the Inequality: $\vert x_1\vert ^\alpha+\vert x_2\vert ^\alpha\geq\Vert X\Vert _2 ^\alpha$, for $\alpha\in (1,2)$

Let $X\in\mathbb{R}^2$, where $X=[x_1,x_2]^T$. If $\alpha\in (1,2)$, then $$\vert x_1\vert ^\alpha+\vert x_2\vert ^\alpha\geq\Vert X\Vert ^\alpha,$$ where $\vert\cdot\vert$ is the absolute value, and $...
0 votes
1 answer
29 views

When substitution works and when it fails (differentiation)

I would like to solve a problem, where I differentiate a function by a fraction, something like this: $$\frac{\partial (x_1^\delta / x_2^\gamma)}{\partial (x_1/x_2)}$$ I was told that such a problem ...
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0 votes
0 answers
9 views

Request for Literature Recommendations on Isotonic Mappings

I am looking for literature recommendations on classical and generalized isotonic mappings of sets. Specifically, I am interested in any scientific and technical literature on this topic that is ...
0 votes
0 answers
15 views

Arithmetic Progression/Sequence | Does it include terms prior to the initial term?

I’m trying to find the right term to describe points along a linear equation f(x) = mx + b, but where the domain is restricted to the set of integers. EDIT: Also the slope and y-intercept are integers....
0 votes
0 answers
50 views

Solve the equation $\sqrt[2021]{\frac{2x+2021}{x+4042}}+\sqrt[2021]{\frac{x+4042}{2x+2021}}=\cos\frac{(2021+x)\pi}{47}+\cos\frac{(2021-x)\pi}{43}$

Solve the equation $$\sqrt[2021]{\dfrac{2x+2021}{x+4042}}+\sqrt[2021]{\dfrac{x+4042}{2x+2021}}=\cos\dfrac{(2021+x)\pi}{47}+\cos\dfrac{(2021-x)\pi}{43}$$ I'd say my observations of the problem are ...
1 vote
1 answer
45 views

How to simplify this equation of acceleration

How does: $$I\frac{a}{R}= -R(m_1g+m_1a)+R(m_2g-m_2a)$$ becomes: $$a = \left(\frac{m_2-m_1}{m_1+m_2+\frac{I}{R^2}} \right)g$$ for context this is the equation for the acceleration of the atwood machine ...
0 votes
1 answer
47 views

Is the trace of a constant simply that constant?

I can't seem to find an answer to this online, basically what I am asking is $$\mathrm{Tr} ((4)_{1\times 1})=4?$$ or in general is $\mathrm{Tr} ((k)_{1\times 1})=k?$ Where $k$ is a constant. I think ...
1 vote
2 answers
112 views

Solve for $x$ and $y$ :

$$bx^3 = 10a^2bx + 3a^3y$$ $$ay^3 = 10ab^2y + 3b^3x$$ I tried putting $bx = p$ and $ay = q$ resulting into system of equations: $$p^3 = 10a^2b^2p + 3a^2b^2q$$ $$q^3 = 10a^2b^2q + 3a^2b^2p$$ ...
0 votes
1 answer
34 views

Need help figuring out a couple of steps behind equation logic

The problem describes the logic behind limits in math. f1 Based on an area under a curve. The overall idea is pretty clear. We split the OX line into several parts with the following coordinates f2 ...
0 votes
1 answer
108 views

If $x$ and $y$ are numbers such that $x + y, x^2 + y^2, x^3 + y^3$ and $x^4 + y^4$ are all integers. Show that $x^n + y^n$ is an integer.

If $x$ and $y$ are numbers such that $x + y, x^2 + y^2, x^3 + y^3$ and $x^4 + y^4$ are all integers. Show that $x^n + y^n$ is an integer for all positive integers $n$. Using induction we have the ...
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3 votes
2 answers
125 views

Is this statement correct : $\frac{dv}{dt}\times dx=\frac{dx}{dt}\times dv$

So I just came across a physics derivation where the treat the $dv$ and $dx$ operators like fractions while I have always heard it's a mistake. But so far what I came up with: $$\begin{align*} \frac{...
0 votes
2 answers
81 views

In how many ways can $2^5\times 3^7$ be factored into three setwise coprime integers?

Let $T$ be the following set of ordered triples, $T=\{(a,b,c):a,b,c\in \mathbb{N}\}$. Find the number of elements in $T$ such that $abc=2^5\times 3^7$ and $\gcd(a,b,c)=1$ My Attempt If $a=2^x;b=3^y;c=...
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2 votes
1 answer
70 views

The Weight Power Mean, how to understand it?

Around this time, I was using the OTIS Excerpt of Evan Chen to prepare (or at least learn competitive math) for the math Olympiad. At the start of the Algebra section, I was introduced to the AM-GM ...
1 vote
0 answers
44 views

maximum with factorials

Let $k>2$. Can we determine $\displaystyle \max_{m\in\mathbb N}\frac1{k10^{m!}}-\frac2{10^{(m+1)!}}$ ? Or at least a non trivial lower bound of this maximum? That is in relation with thread Thanks ...
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0 votes
0 answers
24 views

Simplifying a subindex equation

Consider the following equation $$ \frac{e^{\sum_i \alpha_i x_i}}{\sum_i x_i}=\sum_k\frac{e^{\sum_i \alpha_i y_{i, k}}}{\sum_i y_{i, k}} $$ Is it possible to write $x_i$ as a function of the terms $y_{...
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0 votes
1 answer
43 views

System of parametric equations $E=F=0$ with unknowns $x,y$ [closed]

Solve for $x$ and $y$ where $a$, $b$ are positive reals: $$\frac{x - a}{a ^ 2} + \frac{y - b}{b ^ 2} = \frac1{x - b} - \frac1{y - a} - \frac1{a - b} = 0$$ I've tried to simplify and solve from many ...
0 votes
0 answers
36 views

Determining the asymptotic relationship between two logarithmic functions.

This question is around computer science, but too mathmatichal to ask anywhere else. The log is base 2. These are the two given functions: $g(n) = 2^{\lg^2n}$ $f(n) = \sum_{i=1}^{n^2}\lg\left(i\right)$...
0 votes
2 answers
84 views

How are the domain and range of a quadratic function determined?

How are the domain and range of a quadratic function determined?For example, what are the domain and range of $x^2-4x+10$
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0 votes
1 answer
34 views

Invariant problem starting with the sequence $1,2, ... ,100$

Initially, we are given the sequence $1,2, ... ,100$. Every minute, we erase any two numbers $u$ and $v$ and replace them with the value $uv + u + v$. Clearly, we will be left with just one number ...
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