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.
47,624
questions
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
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 ...
