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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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Disaggregating algebra formula

I have been trying to come up with a way to calculate the individual contribution of each row i. However due to the non-linear relationship implied by the power to 0.5 it is not as straight forward as ...
peho15ae's user avatar
-1 votes
0 answers
21 views

Application of rational functions [closed]

can anybody help with this problem I found on my textbook (no this is not my homework I am just doing some training), I believe is very hard and even the answer provided I the book uses approximation: ...
Federico Ruck's user avatar
0 votes
2 answers
25 views

Can the graph of a rational function cross oblique asymptotes

I have a follow up question from my previous one: I was told that horizontal asymptotes can be crossed by the graphs of rational functions (and I discovered that the vertical ones can't be crossed). ...
Federico Ruck's user avatar
1 vote
4 answers
63 views

For what value of $k$ will $kx^2+x+k=0$ have two real solutions?

The answer key says $-1/2 < k < 1/2$, yet when I tried $k=0$ in the equation, I received only one solution of $x=0$. However, the discriminant with $k =0$, is still greater than $0$. What is ...
thingthingthing123's user avatar
1 vote
2 answers
43 views

If $a+b+c=4, abc=-1, \frac{a}{a^2-3a-1}+ \frac{b}{b^2-3b-1}+ \frac{c}{c^2-3c-1}=\frac{4}{9}$, find $a^2+b^2+c^2$

Given that $a, b, c$ are real numbers, and $a+b+c=4~~~(1)$ $abc=-1 ~~~(2)$ $\displaystyle \frac{a}{a^2-3a-1}+ \frac{b}{b^2-3b-1}+ \frac{c}{c^2-3c-1}=\frac{4}{9} ~~~(3)$ Find the value of $a^2+b^2+c^2....
user1556492's user avatar
0 votes
0 answers
21 views

Combinatorics/ permutations with summations and conditional permutation or rearrangement inequality

This problem was in the Australian Math Olympiad 2024 and was number 3 on the test. I couldn't see the light at the end of the tunnel when solving this problem but rather I could not find a solution ...
Methuka Weerawala's user avatar
0 votes
0 answers
18 views

Local/absolute maximum/minimum for functions whose graph is a straight line

My question: If the graph of a function is a straight line, how would you treat the local and absolute minimum/maximum? Context: This idea came after trying to describe that an absolute minimum (or ...
Kcharliee's user avatar
0 votes
1 answer
42 views

A confusion regarding a specific high school algebric problem [duplicate]

In this equation, $$\frac{a^2 + b^2}{c^2} = \frac{b^2 + c^2}{a^2} = \frac{a^2 + c^2}{b^2} = \frac{1}{k}$$ If we want to find value of k we can calculate algebrically and we would get k=1/2. But the ...
Virender Bhardwaj's user avatar
2 votes
0 answers
48 views

Graphing the rational function $f(x)=\frac{3x}{x^2+x-2}$

Hello all, I was working through this problem and I tried it myself. I found the asymptotes, and due to the rule, the vertical ones are at $x = -2$ and $x = 1$, and the horizontal should be at $y = 0$....
Federico Ruck's user avatar
0 votes
0 answers
60 views

Problem with the derivative of $\sqrt{x}$, because it contains $\sqrt{x}$ itself [closed]

I know that if we take the derivative of $x^{1/2}$ we get $\frac1{2x^{1/2}}$. This is a bit confusing for me, since I wanted to find the slope of $\sqrt{a}$, but it happens so that derivative happens ...
Suchiiswar Paudmanabhan's user avatar
0 votes
0 answers
70 views

Prove $P(x)=x^{2022}+a_{2021}x^{2021}+a_{2020}x^{2020}+\ldots+a_1x+1$ can not have $2022$ real distinct roots

Let $P(x)=x^{2022}+a_{2021}x^{2021}+a_{2020}x^{2020}+\ldots+a_1x+1$ with $a_i \in \mathbb{R}, \forall i = \overline{1,2021}$ such that $|a_{1011}|<2$ and $a_{k}=a_{2022-k},\forall k = \overline{1,...
Lục Trường Phát's user avatar
-4 votes
0 answers
25 views

Is there a way to prove two polynomial $u(x)$ and $v(x)$ are always equal or not for any $A$? [closed]

Is for any $A \geq 0$ in rational field, $u(x)=v(x)$ always? Where, $u(x)= x^6+2x^3+1$ and $v(x)= (1+x^2-2Ax)^3$
vibhu srivastva's user avatar
-1 votes
0 answers
43 views

Understanding why a fraction is the same as a division problem [duplicate]

I am trying to understand why fractions are another way to write division. I haven't seen a satisfying explanation for this yet. I know that division when we say for example a divided by b is like ...
Seeking_The_Truth's user avatar
1 vote
1 answer
95 views

Intuition behind the sum $\sum_{n=1}^{\infty} \frac {n-1}{n!}=1$? [duplicate]

I've been fiddling around with some sums, and got this result: $$\sum_{n=1}^{\infty} \frac {n-1}{n!}=1$$ I've managed to prove this using the Taylor series of e^x, but I'm looking for some Intuition/...
Gilad Peri Glass's user avatar
0 votes
1 answer
49 views

Proof Verification: If $p^k m^2$ is an odd perfect number with special prime $p$, then $p \equiv 1 \pmod 8$ holds.

Let $p^k m^2$ be an odd perfect number with special prime $p$. Since $p \equiv k \equiv 1 \pmod 4$ must hold, then $m^2 - p^k \equiv 0 \pmod 4$. It follows that $$m^2 - p^k = a^2 - b^2 = (a - b)(a + b)...
Jose Arnaldo Bebita Dris's user avatar
2 votes
0 answers
97 views

Prove that sum of $\csc(2^n)$ from $n=3$ to $n=14$ is $0$

I'm having trouble proving the following fact, which was used as a conjecture in another problem: $$\csc (2^3)^\circ + \csc (2^4)^\circ + \csc (2^5)^\circ + \dots + \csc (2^{13})^\circ+\csc (2^{14})^\...
firephoenix16's user avatar
-3 votes
0 answers
24 views

Prove that there exists two disjoint subsets $T, T'\subset S$ such that $ |(\sum_{t\in T}t)-(\sum_{t'\in T'}t')|\le 1. $ [closed]

Suppose that $S$ is a set of 2023 real numbers in the interval $[-10^{100}, 10^{100}]$. Prove that there exists two disjoint subsets $T, T'\subset S$ such that $$ |(\sum_{t\in T}t)-(\sum_{t'\in T'}t')|...
Hermi's user avatar
  • 1,514
0 votes
1 answer
31 views

Analytic solution for equations with non integer polynomial order

I need to quantize the general function $f(x) = a^{-x^b}$ in some arbitrary range from 0 till any m for my program. To do this I calculate a list of ordered xs whoose y values are some set % of the ...
Kryptic Coconut's user avatar
0 votes
1 answer
51 views

Help with AM-GM ineq

We have $abc=1$ and $a^2+b^2+c^2=ab+ac+bc$ with $a,b,c\in\mathbb{R}$. Since $a^2, b^2, c^2\in\mathbb{R^+}$, we can use AM-GM inequality to get $$a^2+b^2+c^2\geq3\cdot\sqrt[3]{(abc)^2}=3$$ Thus, $$a^2+...
Namura's user avatar
  • 125
1 vote
0 answers
85 views

An Optimal way to approach Math Olympiad Number Theory problems [closed]

Q. Positive real numbers a,b satisfy the equation: $\frac{a}{b}$ $\left(\frac{a}{b} + 2\right)$ + $\frac{b}{a}\left(\frac{b}{a} + 2\right) = 2022$ Find $n$ such that: $\sqrt{n}$ = $\sqrt{\frac{a}{b}}...
Pradeep's user avatar
  • 27
0 votes
0 answers
28 views

Positive, Real Roots of Bivariate Polynomial

I have a question regarding lemma 3.1 in this paper. The lemma in question is as follows Consider the function $f(x, \lambda) = ax^3 + bx^2 + cx + d$ where $a > 0$ is fixed but for which the ...
AK4120's user avatar
  • 1
4 votes
4 answers
292 views

Maximum of coefficients of a quadratic equation

Let $f(x)=ax^2+bx+c$ ($a\ne 0$) be a quadratic function. Given a specific $n\in\mathbb N_+$, $f(x)$ satisfies that for all $x\in[-n,n]$, $|f(x)|\leq n$. Find the maximum of $|a|+|b|+|c|$. My initial ...
Flaming's user avatar
  • 107
0 votes
0 answers
56 views

Handling of algebra in differential calculus

3Blue1Brown"Essence of calculus" series called "Derivative formulas through geometry"- 3rd episode of chapter 3 I have considered the area gained to be $dx*(\frac{1}{x}-d(\frac{1}{...
jona173's user avatar
  • 195
3 votes
1 answer
227 views

I wonder where the author used (I) in the above proof. ("Linear Algebra" by Ichiro Satake.)

I am reading "Linear Algebra" (in Japanese) by Ichiro Satake. Theorem 2: The necessary and sufficient condition for $m$ $n$-dimensional vectors $a_j = (a_{ij})$ ($1 \leq j \leq m$) to be ...
佐武五郎's user avatar
  • 1,210
-1 votes
1 answer
68 views

View minus sign as operator or part of the number? How to differentiate? [closed]

I came across this problem,looking at the distributive law "a*(b+c) = ab+ac" / "a*(b-c) = ab-ac". Lets say we have the following term: -4 * (2 - 4) What would you say is c? Is c -4 ...
derflo's user avatar
  • 9
2 votes
4 answers
147 views

Solve $|x|>|x-1|$

Solve $|x|>|x-1|$ $\dfrac{|x|}{|x-1|}>1 \Leftrightarrow \left| \dfrac{x}{x-1} \right| >1$ $\dfrac{x}{x-1} > 1 \tag{1}$ or $-\dfrac{x}{x-1}>1 \Leftrightarrow \dfrac{x}{x-1}<-1 \tag{...
ronald christenkkson's user avatar
3 votes
0 answers
65 views

Solution of equation with unknown under the integral

I have a problem which I have reduced to solving the following equation for the unknown $r_0$: $$ 1/2 = \int_0^D f(r)p(r,r_0)dr $$ where $D \in \mathbb{R}$, and $f$ is continuous density function. $p(...
Ollie's user avatar
  • 103
0 votes
1 answer
60 views

Proof of Unique Factorisation of Polynomials over $\mathbb C$ by Identity Principle

Proposition: Any polynomial $p(x) = a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ can be expressed uniquely as $$ p(x) = a_n(x-r_1)(x-r_2)\cdots(x-r_n), $$ where $r_1, r_2,\ldots, r_n$ (not necessarily ...
prashant sharma's user avatar
0 votes
2 answers
99 views

I don't understand how difference of vectors work {HOMEWORK} [duplicate]

So in the picture we have vectors u and v. Our goal is to find $v−u$ From what I know, the subtraction of vectors is just reversing the direction of the $2^{nd}$ vector & then finding the ...
limaosprey's user avatar
2 votes
1 answer
149 views

Inequalities and averages

Dikshant writes down $2 k+1$ positive integers in a list where $k$ is a positive integer. The integers are not necessarily all distinct, but there are at least three distinct integers in the list. The ...
aiman's user avatar
  • 23
2 votes
1 answer
91 views

What is the Maximum Theoretical Angle a Grand Piano Could be Held At?

Out of curiosity, I wondered why grand pianos have their stand at the length and position that they are made at. I never could find an answer so I decided to try to solve for the maximum angle (B) the ...
Wesley Boudreau's user avatar
0 votes
0 answers
57 views

Summation involving the closest integer to $\sqrt n$ [closed]

Let $f(n)$ be the integer closest to $\sqrt n$. Evaluate $$\sum_{n=1}^\infty\frac{\left(\frac32\right)^{f(n)}+\left(\frac32\right)^{-f(n)}}{\left(\frac32\right)^n}$$ In this question, I was able to ...
cende's user avatar
  • 19
3 votes
0 answers
85 views

Precise Definition of Polynomial [duplicate]

Apologies if this question is too trivial. I am having trouble precisely defining polynomials. All of the definitions I have seen say that expressions of the form $a_nx^n+a_{n-1}x^{n-1}+\ldots+a_1x+...
user985091's user avatar
1 vote
1 answer
87 views

Proving that a set of three quadratic equations in three variables always has a solution [closed]

Given the following system of equations in $s, t, r$: $$ t^2 + s^2 - 2 t s \cos(\theta_1) = a^2 $$ $$ s^2 + r^2 - 2 r s \cos(\theta_2) = a^2 $$ $$ r^2 + t^2 - 2 r t \cos(\theta_3) = a^2 $$ How can I ...
that's what it is's user avatar
1 vote
2 answers
111 views

Solving $ \left|\frac{3x}{7} \right |= 4-x$

I’m trying to solve: $\displaystyle \left|\frac{3x}{7} \right |= 4-x$ Here’s what I’ve tried: $\frac{3x}{7} = 4-x$ (checking for intersections) $x = \frac{14}5$ (this intersection checks out) ...
Mick's user avatar
  • 171
3 votes
0 answers
118 views

AM-GM inequality for non necessary positive numbers

For nonnegative real numbers $x_1,\cdots,x_n$ ($n\geqslant2$), it is well known that : $$n\prod_{i=1}^nx_i\leqslant\sum_{i=1}^nx_i^n\tag{$\star$}$$since this is equivalent to the AM-GM inequality. But ...
Adren's user avatar
  • 7,754
1 vote
1 answer
53 views

Finding a value $n$ such that $\sqrt[n]{x^{x^{2}}} \le x^{\sqrt[n]{x^{2}}}$ is true.

For what value of $n \in \mathbb{N}$ such that the following inequality is true. $$\sqrt[n]{x^{x^{2}}} \le x^{\sqrt[n]{x^{2}}}$$ Where $0<x\le \sqrt[5]{216}$ ATTEMPT: This is my first time tackling ...
JAB's user avatar
  • 325
0 votes
1 answer
48 views

Detrmine the length of the diagonal of a parallelepiped knowing that $a(a^2+3)=b^2+c^2-2a^2+4$

the problem Detrmine the length of the diagonal of a parallelepiped with the dimensions $a,b,c \in(0, \infty)$ which satisfy all of the following equalities: $a(a^2+3)=b^2+c^2-2a^2+4$ $b(b^2+3)=a^2+c^...
IONELA BUCIU's user avatar
3 votes
2 answers
87 views

Solution-verification: Solve $3x^2-6x+4 = 6\{x\}\bigl(\lfloor x\rfloor - \{x\}\bigr)$

the problem Solve in the set of real numbers the following equation $$ 3x^2-6x+4 = 6\{x\}\bigl(\lfloor x\rfloor - \{x\}\bigr), $$ where $\lfloor x\rfloor$ and $\{x\}$ are the whole part and the ...
IONELA BUCIU's user avatar
-4 votes
2 answers
117 views

Find the integers solutions of the equation $x^4+4y^4=3796$ [closed]

the problem Find the integers solutions of the equation $x^4+4y^4=3796$ my idea First thing that came into my mind is that $x^4=4(949-y^4)\Rightarrow 4|x^4 \Rightarrow 2|x$ which means that x is ...
IONELA BUCIU's user avatar
4 votes
2 answers
346 views

Show that $\frac{a^2}{b^2-2b+2024}+ \frac{b^2}{c^2-2c+2024}+ \frac{c^2}{a^2-2a+2024} \geq \frac{3}{2023}$

Let the real numbers $a,b,c \in \mathbb{R}$ with $a+b+c=3$. Show that: $\frac{a^2}{b^2-2b+2024}+ \frac{b^2}{c^2-2c+2024}+ \frac{c^2}{a^2-2a+2024} \geq \frac{3}{2023}$. My idea: First of all, I thought ...
IONELA BUCIU's user avatar
3 votes
3 answers
114 views

Faster way to find self-intersections of the curve parameterized by $(-4t^3-6t^2,-3t^4-4t^3)$

Given is a curve $K$ with $K(t)=\begin{pmatrix}f(t)\\g(t) \end{pmatrix}=\begin{pmatrix}-4t^{3}-6t^{2}\\-3t^{4}-4t^{3} \end{pmatrix}$ and $-1.5 \leq t \leq 0.5$. I want to find the intersection of the ...
garondal's user avatar
  • 889
2 votes
0 answers
104 views

$x_{1}=\frac{1}{x_{1}}+x_{2}=\frac{1}{x_{2}}+x_{3}=\ldots=\frac{1}{x_{n-1}}+x_{n}=\frac{1}{x_{n}}$; prove that $x_{1}=2\cos\frac{\pi}{n+2}$ [closed]

If $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$ are $n\geq 2$ positive real numbers such that $ x_{1}=\frac{1}{x_{1}}+x_{2}=\frac{1}{x_{2}}+x_{3}=\ldots=\frac{1}{x_{n-1}}+x_{n}=\frac{1}{x_{n}}$, prove that $...
Sushil's user avatar
  • 141
3 votes
1 answer
145 views

Prime number as divisor

I was doing a question and I observed a thing that I'm not able to prove, it follows: For any prime number $n>2$ , there must be only one solution $k=n-1%$ (given that $0<k<n-1$ ) to the ...
Someone's user avatar
  • 41
-1 votes
0 answers
69 views

Simplifying $\frac {4x^2 + 6x +2xy + 3y} {4x^2 -9y^2}$

Simplify fully: $$\frac {4x^2 + 6x +2xy + 3y} {4x^2 -9y^2}$$ I think that I need to factorise the top and bottom and cancel out the brackets but the brackets on the top and bottom don't match. Can ...
user22806652's user avatar
1 vote
1 answer
84 views

Shouldn’t $0$ to any power be undefined?

So one of my younger cousins asked me this today. This is the summed up version of what they said. We know that for all real numbers $x^{n-1} = \dfrac{x^n}{x}$, because $x^{n+1} = x^n \cdot x$. So let'...
limaosprey's user avatar
2 votes
2 answers
96 views

$\lim x_n $ in $\frac{1}{x} + \frac{1}{x-1}+\ldots+\frac{1}{x-n}$

Let $x_n \in (0;1)$ be a positive real root of this function: $$f_n(x) = \frac{1}{x} + \frac{1}{x-1}+\ldots+\frac{1}{x-n}$$ with positive integer $n \geq 2$ Find $\lim x_n$ I claimed that $f_n(x) = 0$ ...
Lục Trường Phát's user avatar
-2 votes
1 answer
63 views

What is the product of the solutions to the equation (√10)(x^(log (x))) = x^2? [closed]

I have spent a few hours on this question but I can't seem to grasp it. I have found that taking the log of both sides doesn't give me any progress. The log exponent identity didn't work. I also tried ...
shivank chintalpati's user avatar
2 votes
3 answers
93 views

Quickly finding $(-M)^{ATH}$ from $M+A+T+H=10$, $M-A+T+H=6$, $M+A-T+H=4$, $M+A+T-H=2$

I found this question on social media, from a math account I follow. $$\begin{align} M+A+T+H &=10 \\ M-A+T+H &=\;6 \\ M+A-T+H &=\;4 \\ M+A+T-H &=\;2 \\ (-M)^{ATH} &=\;\text{??} \...
Grey's user avatar
  • 867
1 vote
0 answers
57 views

System of 4 equations with 4 unknowns in Excel: stress and strain evolution during temperature cycles (hysteresis loop)

I am trying to solve the following system of $4$ equations with $4$ unknowns (in red): $$\begin{cases} \color{red}{\gamma_{iv}} + \frac{\color{red}{\tau_{iv}}}{1372} = 4.24 \times 10^{-4} (T_{i + 1} -...
user56288's user avatar

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