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.
47,623
questions
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
333
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 ...
- 889
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 $...
- 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 ...
- 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
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 ...
- 11
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 $f_n(x) = 0$ ...
- 783
-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.
$$\begin{align}
M+A+T+H &=10 \\
M-A+T+H &=\;6 \\
M+A-T+H &=\;4 \\
M+A+T-H &=\;2 \\
(-M)^{ATH} &=\;\text{??}
\...
- 741
1
vote
0
answers
57
views
System of 4 equations with 4 unknowns in Excel: stress and strain evolution during temperature cycles (hysteresis loop)
I am trying to solve the following system of $4$ equations with $4$ unknowns (in red):
$$\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. ...
- 123