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.
46,723
questions
0
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1
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44
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Solving an equation with a shift
My professor offered an extra exercise for us to think about.
The problem is to solve the shifted equation of the form
$$
f(x+ia) = x^2 f(x) \, ,
$$
where $a$ is a constant.
Due to the $x$-dependent ...
- 53
0
votes
0
answers
36
views
What is the image of $\sin(1/x)\cdot(1/x)$, $0<x<1?$
What is the image of $\sin(1/x)\cdot(1/x)$, $0<x<1?$
I just had a question about what the image of $\sin(1/x)\cdot(1/x)$ is for $0<x<1$. Would it not be all the reals, since $\sin(1/x)$ ...
- 1
0
votes
1
answer
38
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prove using induction or in any other way that for all natural number $n≥2$, $3^n>3n+1$
so heres where im at
base $$3^2>3(2)+1$$
$$9>6+1$$
$$9>7$$
which is true
assume $k=n$ $3^k>3n+1$
should be true for n=k+1
$$3^{k+1}>3(k+1)+1$$
$$3^k3>3k+4$$
$$3^k =3k+ \frac{4}{...
- 1
2
votes
2
answers
80
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More elegant way to show $X^2+XY+Y^2=Z^3$, if $X=q^3+3pq^2-p^3$, $Y=-3pq(p+q)$, $Z=p^2+pq+q^2$
I believe I have solved the below problem by just expanding the algebraic terms (I will show this), but I am wondering if there is a more elegant way of making the simplification, or if there is ...
- 627
2
votes
3
answers
92
views
How to prove $(1+\sqrt{2})^n$ is an irrational number for every $n \in \mathbb{N}$?
Let $k,m \in \mathbb{N}$, $a = k + m\sqrt{2}$
I have proved that $a = k + m\sqrt{2}$ is an irrational number.
Now, I am asked to prove that to every $n \in \mathbb{N}$, $(1+\sqrt{2})^n$ is an ...
- 23
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votes
0
answers
47
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In base $12$, divide $15et20$ by $9$ where $e=11$ and $t=10$.
$$
\begin{array}{ rl|ccccccc }
&& & 1&e&e&9&6&.8 \\
\hline
& 9& 1&5&e&t&2&0 \\
& & & 9\\
\hline
&& & 8&e \\
&...
- 1,356
4
votes
1
answer
92
views
Proving $\frac{a+b+c}{a^2b+b^2c+c^2a+9}\ge \frac{abc+6}{abc+27}$ for $a, b, c\ge 0$; $ab + bc + ca = 3$
I came up with the inequality accidentally so there is no original proof so far. It would be great if you can give some useful help to prove it.
Problem. Given non-negative real numbers $a,b,c$ ...
- 3,059
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0
answers
36
views
How to use factor theorem for a polynomial with three binomial factors, i.e. (a+b)(b+c)(a+c)
I am working on a proof and I am completely baffled on how to use factor theorem for more than one binomial factors (as divisors). I find that showing the LHS is equivalent to the RHS is more ...
- 3
0
votes
1
answer
92
views
Function problem, pre-university level
The function $f: R → [-10, +∞) / f(x) = \dfrac{14a-b}{a³(b+5)}x²+(2a-3b)x+(4a-b)$ is even and surjective. If $x_1$ and $x_2$ are their real roots, then $|x_1|+|x_2|$ is equal to? Answer: 9
I have been ...
- 111
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0
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48
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i have a rosace curve with five petals (with r=1+4sin(5θ)), i want to calculate the area of colored part of the pétale using intégrales
This is an image of a rosace curve with five petals. One of the petals is colored in purple, and i do want find the area of the colored surface.
0
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0
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34
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Is my solution to "MacPOW 1141: Capturing 5 integers" correct?
Crossposted from P.SE as it was closed because I sort of forgot the difference between a math puzzle and a math problem
Source: MacPOW 1141
"MacPOW 1141: Capturing 5 integers" states:
For ...
- 2,128
0
votes
1
answer
39
views
Problems on lcm and gcd.
Mario has a rest shift
every $8$ days; Luigi every $24$ days; Paolo every $16$-th
days. Today all three are off.
In how many days will all three be back in
rest shift for the first time?
There are ...
- 6,944
-1
votes
0
answers
77
views
Why is subtraction not associative if addition is? [closed]
We know that:
$$
\forall a,b,c \in \mathbb{R}: (a+b)+c=a+(b+c)
$$
Subtraction can be defined as the addition of the additive inverse of a number.
So $a-b-c$ can also be written as $a + (-b) + (-c)$ ...
- 193
1
vote
0
answers
37
views
How do I ensure that I have not lost/gained any solutions when solving a trig/algebraic equation? [closed]
Sometimes when I try to solve an equation, I need to multiply by a $\cos(x)$ for example to create a common denominator. Does this create a new solution? Why does/doesn't it? When do I know if when I ...
- 45
2
votes
0
answers
59
views
Find the range of the function $f(x)=(\sin^{-1}x)^2-(\cot^{-1}x)^2$
Find range of the function:$f(x)=(\sin^{-1}x)^2-(\cot^{-1}x)^2$
The domain of the function is $-1\leq x \leq 1$
$f(-1)=(\sin^{-1}(-1))^2-(\cot^{-1}(-1))^2=\frac{\pi^2}{4}-\frac{9\pi^2}{16}=-\frac{-5\...
- 8,722
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votes
2
answers
63
views
Does this question have a logical fallacy? [Find the poloynomial satisfying the below condtions.]
This might be silly question, I would ask for your understanding. I got this question from my friend.
$Q)$ Find $f(-1)$ satifying the below condtions from $(1)$ to $(3)$
$(1)$ Dividing the polynomial $...
- 2,214
1
vote
1
answer
29
views
How to algebraically calculate the difference in days for an inverse-proportion or more people=worse problem?
During a drought, 50 people have only $1000 \mathrm{~L}$ of water left. If every person consumes an identical amount of water, the 1000-liter supply would be exhausted in one day. If 40 people were to ...
- 115
-3
votes
0
answers
52
views
Function to fit a set of data points [closed]
I am trying to write an equation to describe a line where the values would be as follows. I am so near yet so far.
I seem to be unable to paste data from Excel without it coming in as a picture.
I ...
- 1
3
votes
0
answers
44
views
How to simplify "median of medians" formula?
Given a nested median formula:
$
\operatorname{med}\big(
\operatorname{med}(a_{1,1},\ldots,a_{1,n}),
\operatorname{med}(a_{2,1},\ldots,a_{2,n}),
\ldots,
\operatorname{med}(a_{k,1},\ldots,a_{k,n})
\big)...
- 10.3k
1
vote
0
answers
50
views
Is the xth root of x always irrational? [closed]
Given x ≠ 0 or 1, I was fiddling with my calculator and discovered the cube root of 3 is irrational (I already knew the square root of 2 is irrational), and that the fourth root of four is irrational. ...
- 19
0
votes
1
answer
85
views
Help with a solution this Fibonacci triangle question
I am currently studying maths at a high school level, I have been given this problem but can’t find any route into it, I have experimented with using Heron’s formula and the expression for the nth ...
- 11
0
votes
2
answers
67
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How to determine argument from Euler’s form of complex numbers?
When writing the Euler’s form of a complex number ($z=r e^{i \varphi}$), we say that $r$ is the magnitude of the complex number, while $\varphi$ is its argument, but the same number can be written in ...
0
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0
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45
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Solution of Two Functions
Given $f(x) = -1 + 5(1.02)^x$ and $g(x) = \ln(3 - x)$, for what value of $x$ does $f(x) = g(x)$? I have been trying to solve this question for quite some time and I always seem to hit a dead end. What ...
- 11
2
votes
0
answers
95
views
Determine all functions $f: \mathbb{Q^+} \to \mathbb{Q^+}$.
Determine all functions $f: \mathbb{Q^+} \to \mathbb{Q^+}$ such that $f(x+1)=f(x)+1$ and $f(x^3) = f(x)^3$ for every $x \in \mathbb{Q^+}$.
I think this one requires me to prove the linearity of the ...
3
votes
2
answers
90
views
Show $\sum_{i=0}^n{i\frac{{n \choose i}i!n(2n-1-i)!}{(2n)!}}=\frac{n}{n+1}$
How can this identity be proved?
$$\sum_{i=0}^n{i\frac{{n \choose i}i!n(2n-1-i)!}{(2n)!}}=\frac{n}{n+1}$$
I encountered this summation in a probability problem, which I was able to solve using ...
user1253495
-2
votes
1
answer
102
views
On the board there are numbers
On the board the numbers $2$, $\sqrt 2$, $\sqrt{3/5}$, $1/3$ are written. Andy will play a game, where each step follows the following sequence:
Andy chooses two numbers on the board, for example $a$ ...
- 13
2
votes
3
answers
85
views
How do we prove that $(x^2 + y^2)^2 + x^2 + y^2 < 1$ is a disk?
How do I see that $(x^2 + y^2)^2 + x^2 + y^2 < 1$ is a disk? I plotted it in wolfram alpha and it looks like a disk, but I don't know how to show it algebraically.
Even if we write in polar ...
- 679
1
vote
1
answer
25
views
Volume of A Solid in 3-space consisting of all points (x,y,z) satisfying the Inequality.
Going through serge langs basic math, the section on Area and Applications.
I've understood that to find the area of an ellipse of the equation
$\frac{x^2}{6} + \frac{y^2}{3}$ = 1
requires one to ...
- 25
-3
votes
1
answer
84
views
Would appreciate help on finding x according to the equations [closed]
I encountered this question while looking through an old textbook (to which I do not have solutions) in the context of learning about the absolute value function at a grade 11 level. I have ...
0
votes
0
answers
20
views
Area Of The Region Bounded By An Ellipse Without Integration
I'm going through serge Lang's Basic Mathematics in order to prepare for university mathematics.
In section 7 on area and applications one must find the area of a region bounded by an ellipse.
$\frac{...
- 25
0
votes
1
answer
52
views
Expanding $\sin(10x)$ with binomial and pascal triangle
I have to derive the formula for $\sin(10x)$ using complex exponent, Binomial formula and Pascal’s triangle to expand the brackets. Here is my solution:
$$
\sin(10x) = \frac{\exp(i10x) + \exp(-i10x)}{...
1
vote
1
answer
81
views
How to prove $(x_1+\frac{1}{x_1})\cdots(x_n+\frac{1}{x_n})\ge (x_1+\frac{1}{x_2})\cdots(x_n+\frac{1}{x_{n-1}})(x_1+\frac{1}{x_n}),n\ge 2$
Recently, when I self-learned Discrete Mathematics and Its Applications 8th by Kenneth Rosen, I did only the even-numbered exercises which the author offers the detailed description than the odd ones ...
- 41
1
vote
1
answer
124
views
Find all the polynomials $f$ that satisfy $f(x^2)+f(x)f(x+1)=0$.
find all the polynomial f that satisfies $f(x^2)+f(x)f(x+1)=0$.
I'm not sure if I'm doing it in the right way and I'm confused.
So I tried to write something like this.
let $z_1,z_2,z_3,...,z_n$ be n ...
0
votes
1
answer
27
views
Zero elements of the exponential of a matrix [closed]
I'm trying to prove that if S is a real symmetric matrix, then the zeroes coefficients of exp(iS) are at the same indexes that those of exp(inS) where n is the dimension of the space.
Any help would ...
- 13
2
votes
1
answer
80
views
Finding the max of a function using $f(x)=L(x)-E(x)$ where $L(x)$ is linear and $E(x)$ is exponential
Take this problem, which was on a younger friend's recent AP Precalc test:
(note that this test was given by an online provider called "AP Classroom". I highly doubt that this was tested ...
- 2,128
3
votes
1
answer
72
views
Inequalities for the solution of $x = (x-a) e^{x+a}$.
Let $a > 0$. The equation
$$(x-a) \, e^{x+a} = x $$
must be solved for $x > 0$. Since the solution does not have a closed form, I would like to obtain bounds for the solution.
Until now, I was ...
- 948
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0
answers
58
views
Differences or ratio of logs?
I have two values that I want to relate with each other based on some quantifications expressed in $\log_2$ scale.
Consider the log2 fold change in disease
$$LFC_{\text{Disease}} = \log_2(A_D / B_D)$$
...
- 331
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votes
0
answers
81
views
If $\sum_{i=1}^n x_i \ge a$, then what can we know about $\sum_{i=1}^n \frac{1}{x_i}$?
Suppose that $$\sum_{i=1}^n x_i \ge a$$
where $a>0$ and $x_i\in (0, b]$ for all $i$. Are there any bounding inequalities we can determine for $$\sum_{i=1}^n \frac{1}{x_i}?$$
I understand that $\...
- 1,511
13
votes
8
answers
4k
views
Why do we sometimes lose solutions when solving equations?
$\forall f:x=y \implies f(x)=f(y)$ which means that any operation can be done on both sides of an equation.
When we solve equations we do one operation after the other:
$$
x=y \implies f_1(x)=f_1(y) \...
- 193
1
vote
1
answer
78
views
Is there a short proof that $-(-n) = n$, when $n$ is a positive integer? [duplicate]
This maybe a silly question, but I was thinking about it.
First we have the set $\mathbb{N}=\{0,1,2,3,...\}$ of natural numbers, then to create $\mathbb{Z}$ we do the following: For each natural ...
- 259
1
vote
0
answers
62
views
How do lines of symmetry change as a graph scales?
Suppose $y=f(x)$ is symmetric in the line $ax+by+c=0$.
Then what is the corresponding line of symmetry for the following ?
$y=df(x)$ and
$y=f(dx)$
I guessed that it'd be
$ax+by/d+c=0$ and
$adx+by+c=...
- 57
1
vote
1
answer
67
views
Is my reasoning right about an extreneous root?
In another post, using this formula for generating Pythagorean triples
\begin{align*}
&A=(2n-1+k)^2-k^2&&=(2n-1)^2+2(2n-1)k\\
&B=2(2n-1+k)k &&=\phantom{(2n-1)^2+{}} 2(2n-1)...
- 6,235
1
vote
3
answers
61
views
How do you write "The sum of all the elements in set A"? [duplicate]
I was in class and the teacher started talking about the mean average. That got me thinking: how would I mathematically represent this? The worded defenition of the mean of set A is "the sum of ...
- 356
-4
votes
0
answers
75
views
Problems with defining $x/0 := 0$, for all $x$? [closed]
The problem with postulating a multiplicative inverse of $0$ in a field is that it implies that $a = b$, for all $a$ and $b$. I tried to discover a similar pathology with the postulate that
$$
x/0 := ...
- 2,021
0
votes
1
answer
66
views
Function $f$ is defined on positive integers, with $f(xy)=f(x)+f(y)$. If $f(40)=20$ and $f(10)=14$, the what is the value of $f(500)$? [closed]
UKMT Hardest Question (2015)
Function is defined on set of positive integers is such that
$$f(xy) = f(x) + f(y)$$
Given that $f(40) = 20$, $f(10) = 14$, what is the value of $f(500)$?
- 23
2
votes
1
answer
69
views
Why won't factoring lead to the same result as expanding when simplifying $(x+y)^2 - y^2 - x^2 + 2xy$?
Hello I was trying to simplify and solve the following equation: $(x+y)^2 - y^2 - x^2 + 2xy$
My first approach was to factor the difference in squares leading to
$$(x+y)^2 - (y+x)(y-x) + 2xy $$
Then ...
- 23
3
votes
4
answers
156
views
How solve the problem $f(x+2)=f(x)+4x+4$ for any $x$
Find $f(2012)$, when $f(2)=0$ and
$$f(x+2)=f(x)+4x+4$$ for any $x$
I tried to find $f(4), f(6), f(8),....$
\begin{align*}
f(4)&=f(2)+4 \cdot 2 +4 \\
f(6)&=f(4)+4 \cdot 4 + 4 = f(2)+4 \cdot (...
0
votes
3
answers
101
views
Help me find a mistake in showing $ \sqrt[4]\frac{7-\cos 4x}{2} > -2\sin x $
Please help me find a mistake in my solution. I have the following inequality:
$$
\sqrt[4]\frac{7-\cos 4x}{2} > -2\sin x
$$
And the following solution:
$$
\frac{7 - \cos 4x}{2} < 16 \sin^4x \\
7 ...
2
votes
3
answers
79
views
Prove that $a_3 \lambda^{3} + a_2 \lambda^{2} + a_1 \lambda + a_0 = 0$ has three real roots
I'm trying to prove that the cubic equation
$a_3 \lambda^{3} + a_2 \lambda^{2} + a_1 \lambda + a_0 = 0$
has three real roots. The coefficients are
$a_3 = - 1 - \sigma - \tau - \chi$
$a_2 = -2 (\sigma +...
- 51
1
vote
3
answers
80
views
Find the minimum value of $\frac{1}{x - y} + \frac{1}{y - z} + \frac{1}{x - z}$ for real numbers $x > y > z$ given $(x - y)(y - z)(x - z) = 17$.
I've tried using AM-GM inequality:
$$
\frac1{x-y}+\frac1{y-z}+\frac1{x-z}\ge3\sqrt[3]{\frac1{(x-y)(y-z)(x-z)}}
$$
Which gives us
$$
\frac1{x-y}+\frac1{y-z}+\frac1{x-z}\ge\frac3{\sqrt[3]{17}}
$$
By ...
- 23