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.
46,725
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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 ...
- 61
2
votes
4
answers
119
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Computing $\sqrt[3]{1\,}$
I know that the answer is always $1$, but they are looking for some way to get to that answer and I don't know what it is. I am not good at english math terms, but maybe it has to do with differential ...
0
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29
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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 $\sin(1/x)$ ...
- 29.9k
2
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2
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75
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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 ...
- 138k
3
votes
4
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153
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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 (...
- 1,627
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1
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34
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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
$$3^{k+1}>3(k+1)+1$$
$$3^k3>3k+4$$
$$3^k =3k+ \frac{4}{...
- 37.6k
2
votes
3
answers
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How to prove $(1+\sqrt{2})^n$ is an irrational number for every $n \in \mathbb{N}$?
Let $k,m \in \mathbb{N}$, $a = k + m\sqrt{2}$
I have proved that $a = k + m\sqrt{2}$ is an irrational number.
Now, I am asked to prove that to every $n \in \mathbb{N}$, $(1+\sqrt{2})^n$ is an ...
- 1,866
4
votes
1
answer
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Proving $\frac{a+b+c}{a^2b+b^2c+c^2a+9}\ge \frac{abc+6}{abc+27}$ for $a, b, c\ge 0$; $ab + bc + ca = 3$
I came up with the inequality accidentally so there is no original proof so far. It would be great if you can give some useful help to prove it.
Problem. Given non-negative real numbers $a,b,c$ ...
- 34.3k
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0
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45
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In base $12$, divide $15et20$ by $9$ where $e=11$ and $t=10$.
$$
\begin{array}{ rl|ccccccc }
&& & 1&e&e&9&6&.8 \\
\hline
& 9& 1&5&e&t&2&0 \\
& & & 9\\
\hline
&& & 8&e \\
&...
- 21.2k
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0
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33
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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
-3
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0
answers
52
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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
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3
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149
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+50
Prove that $\sum\limits_{\mathrm{cyc}}\sqrt{\frac{xy}{z}}+6\sqrt{xyz}\ge \sum\limits_{\mathrm{cyc}} \sqrt{3x^3+6xyz}$
Context. I saw a problem on facebook.It's
Let positive real numbers $x,y,z$ such that $x+y+z=x^2+y^2+z^2$. Prove that$$\sqrt{\frac{xy}{z}}+\sqrt{\frac{yz}{x}}+\sqrt{\frac{zx}{y}}+6\sqrt{xyz}\ge \sqrt{...
- 34.3k
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1
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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 ...
- 148k
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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
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47
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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.
- 33.7k
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+50
Why is only the first (highest) term of the divisor in polynomial long division used to divide?
There is one small matter that has always stumped me with polynomial long division. In the example from the Wikipedia on Polynomial long division, why is the equation only divided by the first/highest ...
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5
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highest product of the numbers that sum to $100$
what is the highest product of the numbers that sum to 100
for example $100 = 1+1+1+1+1+1+1+\ldots+1$ the product of these is just $1^{100} = 1$
$100 = 99 + 1$ the product of these is $99\times 1$
...
- 72.4k
1
vote
0
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50
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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. ...
1
vote
1
answer
138
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Finding minimum $\sqrt{a(2b+1)}+\sqrt{b(2c+1)}+\sqrt{c(2a+1)}$?
Recently, my friend has sent me an excessively hard inequality problem, it's
If $a,b,c\ge 0: ab+bc+ca=3$ then find the minimum $$P=\sqrt{a(2b+1)}+\sqrt{b(2c+1)}+\sqrt{c(2a+1)}.$$
I try to test some ...
- 34.3k
13
votes
8
answers
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Why do we sometimes lose solutions when solving equations?
$\forall f:x=y \implies f(x)=f(y)$ which means that any operation can be done on both sides of an equation.
When we solve equations we do one operation after the other:
$$
x=y \implies f_1(x)=f_1(y) \...
0
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1
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38
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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 ...
- 14.9k
-1
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0
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75
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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 $a + (-b) + (-c)$ ...
- 59.7k
1
vote
0
answers
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How do I ensure that I have not lost/gained any solutions when solving a trig/algebraic equation?
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 ...
- 6,944
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3
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57
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Formula for two overlapping circles, creating three regions with matching area
Suppose we have two circles with radius $r$.
The centers of each circle are separated by a distance $x$, such that $0 < x < r$.
This creates a Venn diagram where you see three distinct regions, ...
- 2,645
0
votes
2
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62
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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 $...
- 269k
2
votes
3
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197
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Least $x$ Such That $\lfloor x^2\rfloor -\lfloor x\rfloor ^2=10$
A friend recently texted me the following:
Compute the least $x$ such that $\lfloor x^2\rfloor -\lfloor x\rfloor ^2=10$.
Is there a way to do what my friend is asking analytically? I graphed it ...
- 181
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0
answers
55
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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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5
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If $a,b,c\ge0:a+b+c=3,$ prove $\sqrt{a+b+b^2}+\sqrt{b+c+c^2}+\sqrt{c+a+a^2}\ge 3\sqrt{3}.$
Problem. Let $a,b,c\ge0:a+b+c=3.$ Prove that $$\sqrt{a+b+b^2}+\sqrt{b+c+c^2}+\sqrt{c+a+a^2}\ge 3\sqrt{3}.$$
Equality holds at $(1,1,1);(0,0,3).$
I tried to use some classical inequality but it seems ...
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1
answer
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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 ...
- 994
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2
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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 ...
- 22.5k
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How to prove this algebraic version of the sine law?
How to solve the following problem from Hall and Knight's Higher Algebra?
Suppose that
\begin{align}
a&=zb+yc,\tag{1}\\
b&=xc+za,\tag{2}\\
c&=ya+xb.\tag{3}
\end{align}
Prove that
$...
- 8,767
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votes
1
answer
83
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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 ...
- 37.7k
-2
votes
1
answer
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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 $a$ ...
- 269k
3
votes
0
answers
44
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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
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votes
1
answer
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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 ...
- 1,511
2
votes
1
answer
90
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Number of solutions for $x ^ y = y ^ x = k$
For a given value of $k \geq 0$, how many solutions $x, y \in \mathbb R$ are there to $x ^ y = y ^ x = k$?
My attempt so far:
There is the "trivial" solution where $x = y$, and the problem ...
- 237
2
votes
1
answer
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What is the linear velocity of the earth in kilometres per hour?
Earth orbits the sun at an average distance of about $150$ million kilometres every $365.2564$ mean solar days, or one sidereal year. What is the linear velocity of the earth in kilometres per hour?
I ...
- 11.5k
2
votes
1
answer
99
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Extraneous solutions in algebraic equations
Consider these 2 Equations , where multiplication by $x$ is involved.
It is known that multiplication by $x$ should introduce the extraneous solution $x=0$ in both cases.
Observe that the second ...
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1
vote
1
answer
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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 ...
- 3,376
0
votes
0
answers
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 ...
- 2,645
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0
answers
95
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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 ...
- 75.2k
3
votes
2
answers
89
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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 ...
- 43.8k
2
votes
3
answers
85
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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 ...
- 15k
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 ...
-3
votes
1
answer
84
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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 ...
- 33.7k
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
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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:
$$
\sin(10x) = \frac{\exp(i10x) + \exp(-i10x)}{...
- 3,884
0
votes
0
answers
81
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If $\sum_{i=1}^n x_i \ge a$, then what can we know about $\sum_{i=1}^n \frac{1}{x_i}$?
Suppose that $$\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
2
votes
3
answers
78
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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 +...
- 34.3k
1
vote
0
answers
130
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Simple algebra in rearring terms
I have a very simple mathematical question, and it is just about algebra which seems very tedious. First, let me state my problem from the beginning:
Let $i$ be an index representing countries ($i = {...
- 133