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.
47,621
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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}{...
- 195
3
votes
1
answer
157
views
I wonder where the author used (I) in the above proof. ("Linear Algebra" by Ichiro Satake.)
I am reading "Linear Algebra" 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 linearly ...
- 1,138
-1
votes
1
answer
55
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View minus sign as operator or part of the number? How to differentiate?
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 ...
- 9
2
votes
4
answers
127
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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{...
3
votes
0
answers
61
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(...
- 103
0
votes
1
answer
51
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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 ...
- 536
0
votes
2
answers
96
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 ...
- 59
2
votes
1
answer
146
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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 ...
- 23
2
votes
1
answer
89
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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 ...
-1
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0
answers
53
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 ...
- 9
3
votes
0
answers
83
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+...
- 346
1
vote
2
answers
108
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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)
...
- 171
3
votes
0
answers
108
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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 ...
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1
vote
1
answer
49
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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 ...
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0
votes
1
answer
47
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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^...
- 748
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 ...
- 748
-4
votes
2
answers
116
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 ...
- 748
4
votes
2
answers
334
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 ...
- 748
3
votes
3
answers
112
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 ...
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2
votes
0
answers
101
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 $...
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3
votes
1
answer
145
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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 ...
- 41
-3
votes
0
answers
90
views
Showing $\frac{\cos(40^\circ)}{2\sin(20^\circ)+\sin(40^\circ)}=\frac{1}{\sqrt{3}}$ [closed]
How would I show this?
$$\frac{\cos(40^\circ)}{2\sin(20^\circ)+\sin(40^\circ)}=\frac{1}{\sqrt{3}}$$
Thanks
-4
votes
0
answers
50
views
Find min degree of required polynomial [closed]
Let $f$ be a polynomial with integer coefficients such that the greatest common divisor of all its coefficients is $1$. For any $n ∈ \mathbb N$, $f(n)$ is a multiple of $85$. Find the smallest ...
-1
votes
0
answers
67
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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 ...
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1
vote
1
answer
79
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'...
- 59
2
votes
2
answers
93
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 $f_n(x) = 0$ ...
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-2
votes
1
answer
62
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 ...
2
votes
3
answers
89
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.
$$\begin{align}
M+A+T+H &=10 \\
M-A+T+H &=\;6 \\
M+A-T+H &=\;4 \\
M+A+T-H &=\;2 \\
(-M)^{ATH} &=\;\text{??}
\...
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1
vote
0
answers
57
views
System of 4 equations with 4 unknowns in Excel: stress and strain evolution during temperature cycles (hysteresis loop)
I am trying to solve the following system of $4$ equations with $4$ unknowns (in red):
$$\begin{cases}
\color{red}{\gamma_{iv}} + \frac{\color{red}{\tau_{iv}}}{1372} = 4.24 \times 10^{-4} (T_{i + 1} -...
- 11
1
vote
1
answer
98
views
Two numbers written on a board get replaced
Question: "Several (at least two) nonzero numbers are written on a board. One
may erase any two numbers, say $a$ and $b$, and then write the numbers
$a+\frac{b}{2}$
and $b−\frac{a}{2}$
instead. ...
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6
votes
4
answers
128
views
$\frac{1}{x} + \frac{1}{x-1} + \ldots + \frac{1}{x-n} = 0$ has only one root $(0;1)$
Prove that: $$\frac{1}{x} + \frac{1}{x-1} + \ldots + \frac{1}{x-n} = 0$$ has only one real root in $(0;1)$ for all positive integer $n>1$
Here is what I tried:
Rewrite the equation as:
$$\frac{(x-1)...
- 783
1
vote
1
answer
73
views
Solving the system $\frac{xy}{ay+bx}=c$, $\frac{xz}{az+cx}=b$, $\frac{yz}{bz+cy}=a$ for $x$, $y$, $z$ [closed]
I came across a question regarding sytems of linear equations. I have tried elimination,substituition and simon's factoring trick etc but still not able to extract x,y,z.
$$
\begin{cases}
\dfrac{xy}{...
- 55
0
votes
2
answers
82
views
A system of equations of power sums for 3 variables
I am currently interested in the following problem:
Find $x, y, z$ such that
$$
\begin{cases}
x + y + z = 2 \\
x^2 + y^2 + z^2 = 6 \\
x^3 + y^3 + z^3 = 8
\end{cases}
$$
I noticed how the solutions ...
- 411
1
vote
2
answers
65
views
Why is interpolating $y=g(x)$ then applying $h(y)$ not equivalent to interpolating $h(g(x))$?
Say I have a table of voltage, current, and resistance values as so.
V [V]
I [A]
R [$\Omega$]
1
2
0.5
3
5
0.6
The V and I columns are measurements, R is a simple calculation from Ohm's law (V=IR).
...
- 13
1
vote
1
answer
86
views
Why do the solutions to $x^2 + 2x + 8\sqrt{x^2 + 2x + 21} - 41 = 0$ change when the equation is manipulated? [duplicate]
Starting with:
$$x^2 + 2x + 8\sqrt{x^2 + 2x + 21} - 41 = 0 \tag{1}$$
If I try to simplify without substitution, by moving the root to the other side, squaring both sides, gathering like terms, I end ...
- 107
-2
votes
0
answers
46
views
How to Rewrite a Polynomial as a Sum of Two Squares Using Maple? [closed]
I need help rewriting the following algebraic expression using Maple
Given the input expression:
$x^2 - 2xy + 2y^2 + 2x - 10y + 17$
The output should be the sum of two squares:
$(x - y + 1)^2 + (y - 4)...
-3
votes
0
answers
51
views
Why I got 2 different result [closed]
enter image description here
Why I got 2 different results?
Where is error?
-1
votes
1
answer
77
views
A person takes 8 minutes to cut a piece of log into 5 pieces. How long would it take to cut it into 10 pieces?
A person takes 8 minutes to cut a piece of log into 5 pieces. How long would it take to cut it into 10 pieces?
One of the solutions which I thought of in the beginning was,
To cut a piece of wood into ...
- 1
-2
votes
2
answers
56
views
Proving $(a^m)^n=a^{mn}$ for all positive natural $m$ and $n$, using double induction [closed]
I need to prove that $$(a^m)^n=a^{mn}$$ for all $m$, $n$ positive naturals (or positive reals actually. But I don't know if a real induction exists, nor would I know how to use it.)
I need to use ...
- 7
-2
votes
0
answers
42
views
How to resolve symbolically "Maximum value of $2$ variable function $f(u,v)=\frac{\left(1-\sqrt{uv}\right)^2}{\frac{1-u^2}{2u}+\frac{1-v^2}{2v}}$"
In my answerto "Maximum value of $2$ variable function $f(u,v)=\frac{\left(1-\sqrt{uv}\right)^2}{\frac{1-u^2}{2u}+\frac{1-v^2}{2v}}$", I arrive at a numerical solution that the maximum is $f(...
- 951
7
votes
1
answer
1k
views
Does there exist a nontrivial "good" set?
We say that a set $A$ is "good" if $A\subseteq\mathbb{R}$, and for every positive integer $n$, if
$a_1,\dots,a_n\in A$, and for every $1\leq i,j\leq n$ with $i\neq j$ we have $a_i\neq a_j$, ...
- 1,148
1
vote
2
answers
63
views
Find a base b in which $\left( 45 \right)_{b}$ and $\left( 55 \right)_{b}$ are squares of consecutive integers
I started with
$$(i) \hspace{5 mm}\left( 55 \right)_{b}-\left(45 \right)_{b}=\left(10 \right)_{b}=\left( b \right)_{10}$$
$$(ii) \hspace{5 mm} \left(x+1 \right)_{b}^2- \left( x \right)_{b}^2=\left( ...
- 57
0
votes
0
answers
41
views
Closed form solution of equation by finding a suitable function
Starting with a sum such as $\sum_{i} b_{i}$, where $b_{i} > 0$ are real numbers for all $i$ under consideration, I have a corresponding vector of real numbers $a_{i} > 0$ for all $i$. I want to ...
- 596
2
votes
2
answers
131
views
Find the maximum of $|a-b|$ if the equation $x^3-x^2+ax-b=0$ has real and positive roots. [closed]
This is an integer type question (round off to nearest integer) stating: "Find the maximum of $|a-b|$ ($a,b \in \mathbb R$) if the equation $x^3-x^2+ax-b=0$ has real and positive roots."
My ...
-4
votes
1
answer
56
views
can we prove $0 < q < 1$ implies $0 < 1- q < 1$ algebraically WITHOUT multiplying by $-1$? [closed]
can we prove $0 < q < 1$ implies $0 < 1- q < 1$ WITHOUT multiplying by $-1$?
Seems like we can't do it. I know it's true, and it's easy to put together word arguments like "the ...
2
votes
1
answer
82
views
If $M:=\sum\limits_{k=1}^{\frac{n(n+1)}{2}}\lfloor\sqrt{2k}\rfloor$ How to Find $\frac{n^3+2n}{M}$?
$$M:=\sum\limits_{k=1}^{\frac{n(n+1)}{2}}\lfloor\sqrt{2k}\rfloor$$
Find $\frac{n^3+2n}{M}$
This problem was on a problem book.
It is easy to find $M$
If $n$ is odd, $\ m=\frac{n+1}{2} $ and $$M= \...
- 6,620
0
votes
1
answer
57
views
Show that $\left| x-y \right| \leq \left| x \right| + \left| y \right|$ for all real numbers $x$ and $y$ [duplicate]
Show that $\left| x-y \right| \leq \left| x \right| + \left| y \right|$ for all real numbers $x$ and $y$
By definition, $-|x| \leq x \leq |x|$ and $-|y|\leq y \leq |y|$.
$\Rightarrow -|x|+|y| \leq x-...
4
votes
7
answers
108
views
Solve $\left| 2x-5 \right| \leq \left|x+4 \right|$
Solve $\left| 2x-5 \right| \leq \left|x+4 \right|$
If both are positive: $2x-5 \leq x+4 \Rightarrow x \leq 9$
If one is negative: $2x-5 \leq -(x+4) \Rightarrow 2x-5 \leq -x-4 \Rightarrow x \leq \frac{...
1
vote
3
answers
78
views
Solve $\left| 3x+2\right| \geq 4$
Solve $\left| 3x+2\right| \geq 4$
If $3x+2 \geq 0$:
$3x+2 \geq 4 \Rightarrow x \geq \dfrac{2}{3}$
If $3x+2 < 0$:
$-(3x+2) \geq 4 \Rightarrow -3x-2 \geq 4 \Rightarrow -x \geq 2 \Rightarrow x \leq -...
2
votes
0
answers
30
views
Is a function whose graph is central symmetric with respect to any point always linear? [duplicate]
If $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfies $f(x+h)-f(x)=f(x)-f(x-h)$ for every $x \in \mathbb{R}, h \geq 0$, must $f$ be of the form $f(x)=ax+b$?
I originally thought to define $f(x)=\frac{1}{...
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