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.
47,263
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Solving $1.20 \cdot 10^6 = \frac{1}{\left (\frac{1}{6 \cdot 10^6}\right ) X}$ for $X$
I realize this rather rudimentary but it has been over a decade since my algebra classes and now I have problem that I can't figure out. I would like someone to walk me through the steps in solving ...
- 23
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2
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Solve $ 11 \cdot 16^{1/(n-1)} = 16^{n/(n-1)} - 10 $
This is probably an easy task for the users here, but I could not solve it.
$$
11 \cdot 16^{1/(n-1)} = 16^{n/(n-1)} - 10
$$
Wolfram Alpha gives the result $ n= 5 $.
What are the steps to solve this?...
- 255
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4
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249
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How to solve systems of three equations?
Either I forgot or never did learn to do it well. I need to solve the following system:
$$9a+3b+c=0$$
$$25a-5b+c=0$$
$$a-b+c=12$$
Google shows me this page with some instructions: http://www....
- 7,191
2
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3
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398
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A five-digit number N....
A five-digit number $N$ is equal to $45$ times the product of its $5$ digits. Find $N$.
Please help. I am not sure how to solve this. I have a feeling it is simple
- 317
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5
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Finding a real solution to a quadratic
I am suppose to find all the solutions to this problem, I think some theorem states that there can only be as many solutions to the problem as the highest degree. I know that calculus reinforces this ...
user138246
2
votes
4
answers
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The number of real roots of $2x^5+8x-7$
Is there a (theoretical) method to tell the number of real roots of $$2x^5+8x-7=0?$$
Note, without using calculator.
- 700
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how to get value of n if the value of n * log n is given? [duplicate]
Possible Duplicate:
How can I solve for $n$ in the equation $n \log n = C$?
how to get value of n if the value of n * log n is given ?
I am stuck with this:
...
- 193
2
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3
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148
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Little help with some algebra
Note: This isn't homework, I'm skipping ahead of class. Please answer all these equations, I'm deathly stuck on them.
Use the substitution method only please. (Find $x$ and $y$.)
\begin{align}
ax\...
- 1,207
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votes
4
answers
133
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how can I solve for $x$?
I am having a bit of trouble solving for x when trying to find $f^{-1}$. I have $$y=\frac{x+5}{x-4}$$ How can I get x on one side? I tried multiplying both sides by the denominator but I am still left ...
- 557
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4
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Solving the cubic polynomial equation $x^3+3x^2-5x-4=0$
How can I solve the cubic polynomial equation $$x^3+3x^2-5x-4=0$$
I simplified it to:
$$x(x^2+3x-5)=4$$
But I don't know where to go from here.
2
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1
answer
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When my teacher gives me a question involving summation notation, do they expect us to calculate it by hand?
Assuming we don't have a calculator that can do summation notation. My class is not up to summation yet, but I'm asking a question involving this concept because I'm not all that experienced using it. ...
- 4,277
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1
answer
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Does $n^2+(n+k)^2+(n+2k)^2+ \ldots +(n+mk)^2$ have a general equation?
Does $n^2+(n+k)^2+(n+2k)^2+\ldots+(n+mk)^2$ have a general formula?
e.g.
$$1^2+2^2+3^2+\ldots+n^2=n(n+1)(2n+1)/6$$
- 13.5k
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3
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248
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What makes $0!$ equal to 1? [duplicate]
Possible Duplicate:
Prove $0! = 1$ from first principles
I don't understand how it's equal to 1. Also, I found that $(-x)!$ is equal to complex $\infty$. How is this so?
- 4,277
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Vectors, find implied dot product?
The task goes as following.
The angle between two vectors $\vec{w}$ and $\vec{r}$ is less than 90 degrees. Vector $\vec{w}$ is given by $\vec{w} = \vec{u} + \vec{v}$ where $\vec{u} \parallel \vec{r}$...
- 1,899
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How to solve $5 - \log_2 (x - 3) = \log_2(x+1)$
Sorry I have to ask such a simple question, my brain is fried after today.
After substituting with a system of equation, I end up with this "simple" logarithmic problem.
$$5 - \log_2 (x - 3) = \...
- 741
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3
answers
244
views
Trying to solve this simple algebra problems: $\frac{5 + 8x}{3 + 2x} = \frac{45 - 8x}{13 - 2x}$
I know it's kind of stupid to ask this question. But I have problems to solve this simple problems. Can someone point me to the right direction? Did I do something wrong in the process or it's a ...
- 35
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4
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Solving $ \log_3 (6x+2) - 2\log_3 (x)=2 $
I am trying to find the value of $x$ ... but I'm absolutely stuck, some hints would be appreciated!
$$ \log_3 (6x+2) - 2\log_3 (x)=2 $$
My work so far:
$$\begin{align*}
\log\left(\dfrac{6x+2}{x^2}\...
- 85
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3
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Find the root for a third degree polynomial?
So far in this course we have not been given any formula for solving third degrees polynomials.$$\frac{1}{3}x^3-2x^2+4x$$
I was thinking about doing it like this
$$x(\frac{1}{3}x^2-2x+4).$$
But that ...
- 1,899
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votes
4
answers
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Range of functions on $\mathbb{R}$
While practicing problems on functions, I am getting a lot of incorrect answers from the given answers. I am a little confused by this as the problems seem simple else I haven't really understood the ...
- 1,321
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2
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Number of equivalence relations on a set
If a set has $n$ elements then what are maximum number of equivalence classes and equivalence relations possible on it?
user6874
2
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2
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299
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Prove in harmonic progression
Prove that if the system of equations $x+2ay + az = 0$,$x+3by+bz = 0$ ,$x+4cy+cz = 0$ has a non-zero solution then a,b,c are in harmonic progression.
I am looking for a suitable approach for this ...
- 22.4k
2
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2
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219
views
Find the inverse of the following function
$g(x) =\frac{3-5x}{5-x}$
I know that I am retarded for not being able to do this on my own. Algebraically you just have to solve for x. Even so, I can't do it.
- 1,899
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2
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98
views
find minimum value of $x^4-x$ without using calculus
as title, is there a way to find the minimum value of $x^4-x$ without using calculus?
By calculus it's easy as $(x^4-x)'=4x^3-1$, so we got $x=\frac1{\sqrt[3]4}$, then we get the min value. But ...
- 5,239
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5
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Given that $A=\text{log}_{16}15$ and $B=\text{log}_{12}18$, find $\text{log}_{25}24$ in terms of A and B.
Given that $A=\text{log}_{16}15$ and $B=\text{log}_{12}18$, find $\text{log}_{25}24$ in terms of $A$ and $B$.
I found that the answer is $\frac{B-5}{-2AB-2A+4B-2}$ but I used a very inefficient steps ...
- 285
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340
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Polynomial problem with vieta formulas
I was working on an algebra problem set and I came across this question:
let
$$
P(x)=\sum_{i=0}^{20} a_i x^i.
$$
Where $a_{20}=1$ and $a_0=1$ and all coefficients are real numbers. The roots of this ...
- 140
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6
answers
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Proof that $\cos(t^2)$ is not periodic
I am trying to prove that $f(t) = \cos(t^2)$ where $t \in \mathbb{R}$ is not periodic.
Suppose this function has a period $B > 0$, i.e. suppose that
$\cos (t^2) = \cos((t+B)^2)$ for every $t$.
Set $...
- 12.6k
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3
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views
What's the domain of $x^{2/6}$? $\mathbb{R}$ or $[0, \infty)$?
Is domain of $f(x) = x^{1/3}$ same as $f(x) = x^{2/6}$?
This is a exam problem in pre-calculus which asked us to write domain of $x^{2/6}$. I think the answer should be $[0, \infty)$ but the teacher ...
- 1,211
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Evaluate $I = \int_0^{10} \lfloor x \rfloor^3 \{x\}\, dx$
Evaluate $I$:
$$I = \int_0^{10} \lfloor x \rfloor^3 \{x\}\,dx.$$
[$\{.\}$ represents the fractional part of $x$]
[ $\lfloor x \rfloor$ represents the greatest integer function / floor function of $x$]
...
- 2,484
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2
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196
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Is the equal symbol in an infinite series misleading notation?
The infinite series in this notation: $$\frac12+\frac14+\frac18+\dots = 1$$
is nothing more than the limit of the partial sums in:
$$\sum_{n=1}^\infty\frac{1}{2^{n}}$$
The initial notation implies to ...
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5
answers
160
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Let $a^2+b^2=1,c^2+d^2=1$ and $ac+bd=0.$ Prove that $a^2+c^2=1, b^2+d^2=1$ and $ab+cd=0.$
Let $a^2+b^2=1,c^2+d^2=1$ and $ac+bd=0.$ Prove that $a^2+c^2=1, b^2+d^2=1$ and $ab+cd=0.$
My solution goes like this:
We have, $$(a^2+b^2)(c^2+d^2)=1=(ac+bd)^2+1\implies a^2c^2+a^2d^2+b^2c^2+b^2d^2=a^...
- 2,614
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2
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How to recognize a parabola from an algebraic expression?
Context. I worked on an intermediate-value-theorem problem that asked whether $f(x) = x^{10} - 10x^2 + 5$ had a root in $(0, 2)$. I solved the problem by computing $f(0)$ and $f(2)$, both of which ...
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3
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Finding the total number of objects(stones).
I have $N$ stones. Then the stones are arranged in ascending order of weights. If I remove three stones that are heaviest, then the total weight of the stones decreases by $35$%. Now if I remove the ...
- 399
2
votes
2
answers
227
views
Number of handshakes in a party.
After a dinner every person in a party shakes hands with the others. Total number of hand shakes is $105$. Then the number of persons attended that party is:
The correct answer was given as $15$.
...
- 529
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4
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Is it possible to convert a complex solution to a real solution? [closed]
I am a beginner, when it comes to higher level math, so bear with me, But I have been learning about imaginary numbers, and I found something quite concerning.
Imaginary numbers are used to resolve ...
- 131
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5
answers
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Finding second derivative of $x^{x^x}$
If $f(x) = x^{x^x} $, then find $f''(1)$.
My attempt:Now,
$\begin{align}y =x^{x^x} \\&\implies ln(y) = x^x \cdot ln(x)
\\&\implies ln(ln(y))= x\cdot ln(x)+ln(ln(x))\\&\implies\frac{y'}{y\...
2
votes
5
answers
97
views
How to decompose $\frac{1}{(1 + x)(1 - x)^2}$ into partial fractions
Good Day.
I was trying to decompose $$\frac{1}{(1 + x)(1 - x)^2}$$ into partial fractions.
$$\frac{1}{(1 + x)(1 - x)^2} = \frac{A}{1 + x} + \frac{B}{(1 - x)^2}$$
$$1 = A(1 - x)^ 2 + B(1 + x)$$
...
- 1,858
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votes
3
answers
201
views
Squaring $\sqrt{(x-2)^{2}+(y-3)^{2}}=\frac{|5x+6y+5|}{\sqrt{5^{2}+6^{2}}}$ does not lose information?
Scenario 1:
$$y=3x+5\tag{1}$$
$$y^{2}=\left(3x+5\right)^{2}\tag{2}$$
When both sides are squared in $(1)$, we lose some information and get $(2)$: the graphs of $(1)$ and $(2)$ aren't the same.
...
- 3,176
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3
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139
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Find all $f(x)$ such that $x(f(x+1)-f(x))=f(x)$
The problem
Find all $f(x): \mathbb{R} \to \mathbb{R} $ such that $x(f(x+1)-f(x))=f(x)$ and $|f(x) -f(y)| \le |x-y|, \forall x,y \in \mathbb{R}$
My approach
Obviously $f(x)=x$ is one solution, I ...
- 1,288
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votes
3
answers
870
views
How to evaluate $\lim\limits_{x\to \infty}\frac {2x^4-3x^2+1}{6x^4+x^3-3x}$?
Evaluate: $\lim\limits_{x\to \infty}\frac {2x^4-3x^2+1}{6x^4+x^3-3x}$.
I've just started learning limits and calculus and this is an exercise problem from my textbook.
To solve the problem, I tried ...
- 3,956
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votes
3
answers
144
views
If $2\sin\theta+\cos\theta=\sqrt3$, what is the value of $\tan^2\theta+4\tan\theta$?
If $2\sin\theta+\cos\theta=\sqrt3$, what is the value of $\tan^2\theta+4\tan\theta$ ?
$1)1\qquad\qquad2)2\qquad\qquad3)3\qquad\qquad4)5$
First I tried plugging in some values for $\theta$ like $0,\...
- 6,475
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votes
4
answers
90
views
If for the real numbers $a,b(a\ne b)$ it is true that $a^2=2b+15$ and $b^2=2a+15$, then what is the value of the product $ab$?
If for the real numbers $a,b(a\ne b)$ it is true that $a^2=2b+15$ and $b^2=2a+15$, then what is the value of the product $ab$?
I tried to solve it as follows:
I state that $p=ab$
$p^2=(2b+15)(2a+15)$
$...
- 1,197
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3
answers
79
views
$(x^2-9)^{(3x+5)}=(x-3)^{(x-1)}(x+3)^{(x-1)}$ What is $x$?
$(x^2-9)^{(3x+5)}=(x-3)^{(x-1)}(x+3)^{(x-1)}$
$(x^2-9)^{(3x+5)}=(x^2-9)^{(x-1)}$
$3x+5=x-1$
$x=-3$
But when I try to use WolfarmAlpha the integer solution is $3$ instead of $-3$. The numerical ...
2
votes
2
answers
384
views
SMO problem: Sequence and series.
Problem on Series and Sequences (SMO Test):
For each positive integer $n \ge 1$ , we define the recursive relation given by $a_{n+1}=\cfrac{1}{1+a_n}$.
Suppose that $a_1=a_{2012}$.Find the sum of ...
user880107
2
votes
4
answers
129
views
How do I solve this equation over integers: $1=91p+74q$ [duplicate]
Like in title. I need to solve this equation: $$1=91p+74q$$ with $p, q$ being integers. I brute forced the solution to be $(p,q)=(-13,16)$, but I'd love to know how to solve it without just mashing ...
- 85
2
votes
5
answers
2k
views
How to prove that $a^3 + b^3 \geq a^2b + ab^2$?
Um I am solving problems in Arthur Engels book "Problem Solving Strategies". I was doing a problem from inequalities chapter, and I stumbled across a problem which I managed to condense and ...
- 436
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votes
2
answers
463
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If $\frac{(a-b)(c-d)}{(b-c)(d-a)} = \frac{2016}{2017}$ , find $\frac{(a-c)(b-d)}{(a-b)(c-d)}$ .
If $\frac{(a-b)(c-d)}{(b-c)(d-a)} = \frac{2016}{2017}$ , find $\frac{(a-c)(b-d)}{(a-b)(c-d)}$ .
What I Tried :- First I thought for a moment and found out that I can write this :
$$\frac{(a-c)(b-d)}{(...
- 4,223
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votes
5
answers
1k
views
Find the maximum value of $\sqrt{x - 144} + \sqrt{722 - x}$
Find the maximum value of $\sqrt{x - 144} + \sqrt{722 - x} .$
What I Tried :- Let me tell you first . What I Tried is absolutely silly , but you may check for it .
I thought AM-GM would do the trick ...
- 4,223
2
votes
2
answers
999
views
If $a$, $b$, $c$ are the roots of $x^3-6x^2+3x+1=0$, find all possible values of $a^2b+b^2c+c^2a$
Let $a$, $b$, $c$ be the roots of
$$x^3 - 6x^2 + 3x + 1 = 0$$ Find all possible values of $a^2 b + b^2 c + c^2 a$. Express all the possible values, in commas.
I've already tried to bash out all the ...
- 541
2
votes
5
answers
184
views
If $x, y, z\in\mathbb R^+ $ and $x^3+y^3=z^3,$ then prove that $x^2+y^2-z^2>6(z-x) (z-y). $
I made several unsuccessful attempts. Still couldn't think of a proper way to prove the inequality. Please suggest how to approach this problem. Thanks in advance.
EDIT 1. My approach (that I was ...
2
votes
2
answers
582
views
$P(x)=P(-x)$ holds for all values of $x$ ,two conditions
I was doing a question on polynomials , where it was found that
$P(x)=P(-x)$ in the interval $[-\sqrt2,\sqrt 2 ]$
It was then concluded that $P(x)=P(-x)$ holds for all values of $x$ ,"since it is a ...
- 59