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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.

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Is there an exact answer to this problem?

enter to see the problem To translated from Thai English as I could If y is a real number that satifies y^{5} + y - 1 = 0 then find y^{3} + y^{2} +1 I tried to solve it by substitution but still can'...
PeanutSnack's user avatar
1 vote
0 answers
20 views

Inscribing two circles and an ellipse in a square

A square of given side length $S$ is to inscribe two circles and an ellipse as shown in the figure below. If the radius of circle $(1)$ is given, determine the center and radius of circle $(2)$, then ...
c'est pas normale's user avatar
0 votes
2 answers
61 views

Get rid of irrationality in the denominator $\frac{9-40\sqrt[3]{6}-6\sqrt[3]{36}}{1-\sqrt[3]{6}-3\sqrt[3]{36}}$

Get rid of irrationality in the denominator $$\cfrac{9-40\sqrt[3]{6}-6\sqrt[3]{36}}{1-\sqrt[3]{6}-3\sqrt[3]{36}}$$ $$\left ( 1-\sqrt[3]{6}-3\sqrt[3]{36} \right )\left ( 1+\sqrt[3]{6}-3\sqrt[3]{36} \...
Dmitry's user avatar
  • 1,362
-4 votes
0 answers
48 views

Assistance in expression expansion. [closed]

I wonder if someone could shed light on this: Some instances of the expression $(A+B\sqrt{C})^D$ , in which $A,B,C$, and $D$ represent distinct single non-zero digits, may be expanded to give $E + F\...
Paul Woolley's user avatar
0 votes
0 answers
28 views

Repeating sum of no. of terms in a sequence

In the sequence $a_1, a_2, a_3, \ldots, a_n$, the sum of any three consecutive terms is 40; if the third term is $10$ and the eighth term is $8$; find the $2013$th term. I see that if there is a ...
Shiv's user avatar
  • 231
1 vote
1 answer
56 views

A conceptual misunderstanding in limits and geometric series

So, it starts off like this: My friends told me that the floor function of 0.9 repeating (a.k.a. 0.99999.... ) is 0, which is factually untrue since 0.9 repeating is known and proven to be exactly ...
32 Bit's user avatar
  • 13
1 vote
0 answers
37 views

How to see that "if the leading coefficient is 1 or -1, then the rational zeros must be factors of the constant term" in a polynomial

If p/q is a rational zero, in lowest terms, of a polynomial P, then to find its rational zero: $$a_n(\frac{p}{q})^n + a_{n-1}(\frac{p}{q})^{n-1} + \cdots + a_1(\frac{p}{q}) + a_0 = 0 \therefore$$ $$\...
Guilherme Cintra's user avatar
1 vote
1 answer
96 views

Nit picking at whether these two functions are the same

I have a simple and dumb question: are these two functions the same? $$ f(x)=x $$ and $$ g(x)=\frac{x^2}{x} $$ Obviously f and g are the same since you can simplify by x but are they really? $f$ is ...
Skepta's user avatar
  • 121
-2 votes
0 answers
37 views

Is it possible to calculate power of binary number like $a^b$ where $a$ is binary number and $b$ is decimal number?

Is it possible to calculate power of binary number like $a^b$ where $a$ is binary number and $b$ is decimal number? I mean, is there a formula for it?
Lelianeli's user avatar
2 votes
1 answer
23 views

If $9^{x+1} + (a^2 - 4a - 2)3^x + 1> 0$ for all $x ∈ R$, then which of the following is true?

If $9^{x+1} + (a^2 - 4a - 2)3^x + 1> 0$ for all $x ∈ R$, then which of the following is true? The options given are as follows : a) $a ∈ R$ b) $a ∈ R^+$ c) $a ∈ [1,∞)$ d) $a ∈ R -$ {2} The answer ...
1025's user avatar
  • 39
-3 votes
1 answer
62 views

$x=p+\frac{1}{p}$ , $y=q+\frac{1}{q} $ and $z=\frac{p}{q}+\frac{q}{p}$. Then find the value of $x^2+y^2+z^2-xyz$. [closed]

$x=p+\frac{1}{p}$ , $y=q+\frac{1}{q} $ and $z=\frac{p}{q}+\frac{q}{p}$. Then find the value of $x^2+y^2+z^2-xyz$. In this question I have found the answer by putting random values of p and q directly. ...
Shlok Bansode's user avatar
-1 votes
1 answer
13 views

Conversion of a result number to a specified unit of measure

It's a very simple question, but I need to know how to convert numbers, because in several cases I ran into this topic and felt unsure about this. So for example we're counting a unit measure of time, ...
Norbert Kts's user avatar
11 votes
9 answers
2k views

Doubts regarding absolute value

My fiancée is a teacher in a secondary school. She asked me a question connected to absolute value that I can't answer. Let's consider the following problem $$ \lvert x - 2 \rvert < \lvert x + 4 \...
Hendrra's user avatar
  • 2,990
-1 votes
1 answer
100 views

Why is this complicated expresssion equal to 1? [duplicate]

In solving a physics problem, the given answer and the one I obtained differ by the factor $$k(x)=\frac{1+3x^2-x\sqrt{5+9x^2}+x\sqrt{5+13x^2-4x\sqrt{5+9x^2}}}{1+x^2}$$ Graphing it in desmos showed, ...
Vulgar Mechanick's user avatar
11 votes
2 answers
257 views

$x^x = (x-1)^{x+1}$

Background: I was trying to estimate the size of $21^{21}$ for some problem and decided to use $20^{22}$ as hopefully a rough approximate ($20^{22} = 2^{22} \cdot 10^{22} \approx 10^{28}$). But then I ...
mpear617's user avatar
  • 318
1 vote
4 answers
890 views

Why roots aren't the inverse of exponentiation but logarithms?

I think it's easy to see it when we look at the inverse of the function "$f(x) = a^x$" but I wonder if there's other way to look at it besides just analyzing the function. I was taught my ...
pingu's user avatar
  • 21
0 votes
2 answers
50 views

Question about the spread of a disease. Similar to compound interest or continous compounding.

Suppose there are 5 people sick on the first week of an epidemic. During next week, they each make 4 people sick. How many people will be affected during tenth week of the epidemic? Note: Many are ...
Lucas Town 's user avatar
5 votes
2 answers
349 views

Find the radius of the red circle

Four circles are arranged as shown in the figure below. They're numbered from $1$ to $4$. If the diameter of circle $C_2$ is equal to $9$, and $ PT = 6 , QT = 3 \sqrt{5} $. It is also given that $UV ...
c'est pas normale's user avatar
3 votes
2 answers
93 views

Is there a way to show that $5a(a+1)\over 3a+4$ $\notin \mathbb{N}$ when $a\in\mathbb N\setminus\{2\}$? [duplicate]

Is there a way to show that $5a(a+1)\over 3a+4$ $\notin \mathbb{N}$ for $a\in \mathbb{N}$ (except when $a=2$)? The expression has a few different forms, but I don't see how to show this. Any hints are ...
ddswsd's user avatar
  • 1,379
4 votes
2 answers
185 views

Number of animal problems [closed]

My child is trying to solve this problem: For every $3$ chickens on a farm, there were $4$ ducks. For every $9$ chickens, there were $2$ cows. After $15$ chickens were sold, $4/11$ of the remaining ...
user1326484's user avatar
0 votes
1 answer
64 views

Forming an O.D.E for $y=A\sin(Bx)+B\cos(Bx)$

Forming an O.D.E for $y=A\sin(Bx)+B\cos(Bx)$, Where $A$ and $B$ are the arbitrary constants. My efforts: We have $$y'=AB\cos(Bx)-B^2\sin(Bx)$$ Also $$y''=-AB^2\sin(Bx)-B^3\cos(Bx)=-B^2y$$ Now I am ...
LifeIsMath's user avatar
-3 votes
0 answers
36 views

Set of all x such that 4<x≤7•5 [closed]

Draw each of the following subsets of R for those that are given in terms of absolute values write an alternative description that does not the absolute value.
Sawera Chachar's user avatar
0 votes
0 answers
38 views

What is the minimum value of the expression? [duplicate]

Given reals $x_i \ge 0$ for $i=1,2,3,...,2n$ and $\sum_{i=1}^{2n}x_i=1$ find the minimum value of $\sum_{i=1}^{n}x_i+\sum_{i=n+1}^{2n}{x_i}^2$. I tried the case $n=1$ and got the value $\frac{3}{4}$. ...
Suprativ Mondal's user avatar
4 votes
2 answers
116 views

Understanding this ratio trick

Two numbers are in the ratio $3:5.$ If $9$ is subtracted from each, they become in the ratio $12:23.$ Find the smaller number. Solution Let's denote the two numbers as $3x$ and $5x$. According to the ...
vj Maths's user avatar
-1 votes
0 answers
57 views

Solve the question [closed]

Each of the numbers $x_1,\dots,x_{101}$ is $\pm1$. What is the smallest positive value of $$\sum_{1\le i\le j\le101}x_ix_j\ ?$$ Try to sove the question
Prathamesh Laddhad's user avatar
1 vote
1 answer
63 views

Formula for Sum of Numbers with Consecutive Digits [closed]

Let's say we have a number $n$ that consists of $b$ digits, all of which are the same digit $x$. Then we have another number $m$ that has $b - 1$ digits, also all the same digit $x$. How would you ...
Sak's user avatar
  • 21
0 votes
2 answers
58 views

Find the length of segment $AD$ in this trapezoid [closed]

$ABCD$ is a trapezoid with $AB \parallel CD$ and $AB \perp BC$. In addition, the two diagonals $AC$ and $BD$ satisfy $ AC \perp BD$. Segment $BC = 3$. And finally, the area of the blue triangle is ...
c'est pas normale's user avatar
10 votes
3 answers
1k views

Is there is way to determine if the n-th roots of a polynomial is a polynomial?

I was this problem: $$\int\frac{dx}{\sqrt{x^4+2x^3+3x^2+2x+1}}$$ I solved this question because I just knew that $(1+x+x^2)^2=x^4+2x^3+3x^2+2x+1$ but this made me wonder is there is a way to know if ...
pie's user avatar
  • 5,645
8 votes
5 answers
266 views

Manipulating Algebraic Equations

I'm sorry if this question (or possibly multiple depending on how long this I intend for this to be) is a little too elementary, but I've been seriously struggling with this for the past week. I ...
Lucas's user avatar
  • 81
1 vote
1 answer
40 views

Proof that two matrices are row-equivalent iff they have the same nullspace

The matrices are both of size m x n over some field F, obviously. The first direction of this proposition is clear enough, however the opposite direction (same nullspace -> row-equivalence) is ...
Blabla's user avatar
  • 179
1 vote
2 answers
73 views

Find $x$ in this concyclic quadrilateral

$ABCD$ is a concyclic quadrilateral, with $\angle A = 60^\circ$, and $ AB = 10, BC = x , CD = x+2 , DA = x+4 $. Find $x$. My attempt: Using the vector method, we can express the horizontal and ...
c'est pas normale's user avatar
0 votes
1 answer
64 views

Formula for Root Series

Given the series: $S_n=\sqrt{c}+\sqrt{c-1}+\sqrt{c-2}+\sqrt{c-3}+...+\sqrt{c-n}$ in which $c$ can be any value but independent from n. Is there a formula for these kinds of series? Excluding the ...
Anthony's user avatar
0 votes
2 answers
103 views

Calculating circle offset for each angle when reference point is not circle center

I try to calculate offset change at outer rim of a circle for each angle with respect to C1 but can not figure out. its is easy on the paper with geometry but very difficult to formulate it. The ...
ebbac44's user avatar
  • 11
-2 votes
0 answers
42 views

Proving a tough inequality [closed]

enter image description here Take no heed of parts i) and ii) and instead directly to part iii) how to prove that? I could start by breaking up the squares algebraically, but other than that, I haven'...
Enshu Liu Dongfang's user avatar
-5 votes
0 answers
47 views

Trouble in understanding the series $\sum_{n = 1}^\infty \left[\frac{4}{n(n + 1)} - \frac{1}{3^n} \right]$ [closed]

Find the sum of the series $$\sum_{n = 1}^\infty \left[\dfrac{4}{n(n + 1)} - \dfrac{1}{3^n} \right]. $$ I can't seem to end on a finite sum for this series when you split it into twos because the $-1/...
Kermitheweeb's user avatar
3 votes
3 answers
139 views

Find $x$ in this quadrilateral

A quadrialteral $ABCD$ has $AB = 10$, $\angle A = 50^\circ, \angle B = 120^\circ$, $ BC = x , CD = x + 2 , AD = x + 4 $. Find $x$. My attempt: Applying the law of cosines to $\triangle DAC$ and $\...
c'est pas normale's user avatar
3 votes
3 answers
150 views

Find the exact value of $\DeclareMathOperator{\cosec}{cosec} \cosec(10^\circ) + \cosec(50^\circ) - \cosec(70^\circ)$ [duplicate]

Find the exact value of $\cosec(10^\circ) + \cosec(50^\circ) - \cosec(70^\circ)$. The equation can be written as $$ \cosec(x) + \cosec(60^\circ-x) - \cosec(60^\circ+x), $$ where $x = 10^\circ$. I ...
Hitesh's user avatar
  • 67
0 votes
1 answer
49 views

Relationship between roots and coefficients of a cubic

Let $f(x) =x^3-px+q,p>0,q>0$ and all the zeroes of $f(x) $ are real. Prove that if $\alpha$ be the root with least absolute value then $|\alpha|$ lies in the interval $(q/p, 3q/2p) $. I have ...
YBR's user avatar
  • 75
0 votes
1 answer
96 views

Floor function with factorials

$x$ is an integer satisfying: $$\left[\frac x{1!}\right]+\left[\frac x{2!}\right]+\left[\frac x{3!}\right] + \cdots +\left[\frac x{10!}\right]=1001.$$ Find largest prime divisor of $x$, where $[~]$ ...
Quark's user avatar
  • 3
-3 votes
0 answers
69 views

Solving GIF questions [closed]

The value of [1+1/√3+1/√5+.......+1/√121] is equal to? A)10 B)11 C)9 D)12 Where [.] denotes greatest integer function How to proceed with these questions? I am thinking with approximation something ...
Loci's user avatar
  • 7
1 vote
1 answer
24 views

Prove the gambler can expect to play forever in the gambler’s ruin scenario

In mcs.pdf, Problem 21.2 says: In a gambler’s ruin scenario, the gambler makes independent $1$ bets, where the probability of winning a bet is p and of losing is $q ::= 1-p$. The gambler keeps ...
An5Drama's user avatar
  • 390
-1 votes
0 answers
26 views

Trigonometric formulas derivation? [closed]

We know that how other formulas like quadratic formula, distance formula, section formula are derived but what about trigonometry formulas? We are just told to memorize their values. Pardon me for ...
Payal Payal's user avatar
0 votes
1 answer
80 views

Is it possible for a quadratic function to not have $y$-intercept?

I know quadratic functions may have up to two $x$-intercepts, but can they have no $y$-intercepts? Edit: I mean quadratic functions which output parabolas (i.e. $f(x)=ax^2+bx+c$ where $a\neq0$). So ...
Vee's user avatar
  • 11
-1 votes
0 answers
89 views

Decide whether the polynomial $ x^{154} - 2x - 3 $ is divisible without a remainder by the polynomial [duplicate]

I have a question regarding polynomial division. I had this problem at one of my test. Decide whether the polynomial $ x^{154} - 2x - 3 $ is divisible without a remainder by the polynomial: a) $ x + 1 ...
peterparker321's user avatar
2 votes
2 answers
174 views

Find out $\int_{0}^{1} \frac{\ln(-x^2+x+1)}{x(1-x)} \rm{d}x$ without Feynman .

Find out $$\int_{0}^{1} \frac{\ln(-x^2+x+1)}{x(1-x)} \rm{d}x$$. They say when it comes to logarithms in integrals first thing to check if $[x → \frac{1}{x}]$ works or not . My try : $$I = \int_{0}^{1} ...
Ash_Blanc's user avatar
  • 847
0 votes
3 answers
88 views

Difficult algebra problem related to finding a specific value in terms of "$c$" or "$b$"

If $\frac{a}{b}=\frac{b}{c}=\frac{c}{a}$ and $ax^2+bx+c=0$ then the value of $b^2x^4+b^2c^2x^2+2b^2cx^3$ is A) $c^4$ $\quad$B) $-c$ $\quad$C) $-c^2$ $\quad$D) $b^4$ Can anyone please help me with this ...
MIND FORGE NEXUS's user avatar
0 votes
1 answer
60 views

Suppose that $-1\leq ax^2 + bx + c\leq 1$ for $-1\leq x\leq 1$, where $a, b$ and $c$ are real numbers, prove that $-4\leq 2ax + b\leq 4$. [duplicate]

I see that for $x = 0, |c|\leq 1$, for $x = -1, |a - b + c|\leq 1$ and for $x = 1, |a + b + c|\leq 1$. Thus, $|2a + 2c|\leq 2\Rightarrow |2a|< 4$ and $|b|\leq 1$. This, gives me $|2a + b|\leq 5$, ...
Shiv's user avatar
  • 231
1 vote
1 answer
53 views

In an isosceles triangle with base $a$ and congruent side $b$ the vertex angle is equal to $20°$. Prove that $a^3 + b^3 = 3ab^2$.

I was trying to solve this problem: In an isosceles triangle with base $a$ and congruent side $b$ the vertex angle is equal to $20°$. Prove that $a^3 + b^3 = 3ab^2$. After a long time of thinking ...
pie's user avatar
  • 5,645
-6 votes
2 answers
83 views

Integrate $ \int_{} \frac{\sin(x-a)}{\sin(x-b)} {\rm{d}}x$ smartly than rest?

Integrate $ \int_{} \frac{\sin(x-a)}{\sin(x-b)} {\rm{d}}x$ I am expecting better new elegant approaches for this simple integral and lets see the no. of other ways to solve it. My try : $$I = \int_{}...
Ash_Blanc's user avatar
  • 847
1 vote
1 answer
62 views

Integrate $ \int_{} 1 + \tan x\tan(x+\theta) {\rm{d}}x$ quickly .

Integrate $ \int_{} 1 + \tan x\tan(x+\theta) {\rm{d}}x$ I am expecting better new elegant approaches for this simple integral and let us see how shorter it can be. My try : $$I = \int_{} 1 + \tan x\...
Ash_Blanc's user avatar
  • 847

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