Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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2answers
21 views

Logic type problem.

A school has 90 children.During the day each child attends 4 classes.Each class has 15 children and 1 teacher.During the day each teacher has 3 classes.What is the smalles number of teachers the ...
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1answer
22 views

Non-homogenous System where did I go wrong?

Solve the system $\vec{x^{'}}=\begin{pmatrix}2 & -5\\1 & -2 \end{pmatrix}\vec{x}+ \begin{pmatrix} -\cos t\\ \sin t \end{pmatrix}$ The Eigenvalues are $(2-\lambda)(-2-\lambda)+5=0 \implies ...
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1answer
26 views

How many originally… Counting Backwards

Three pirates were stranded on a desert island just outside of Pittsburgh. They came upon a treasure chest, which they opened and found to be full of oranges. Unfortunately they couldn't decide how to ...
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1answer
22 views

Matrices word problem?

Julie and Bill are waiters at the Ogling Ogre Convention Center, which is well-known for serving the most deliciously disgusting meals to its guests. One of their tasks was to count the 2-eyed and ...
3
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0answers
20 views

Order of operations involving mod (%)

I was taking a programming test last night that had a math equation that simplified to 11 % 2 * 3, no () or likewise. When I compute it, being taught modulous occurs at the same level of ...
3
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5answers
412 views

How to solve for a variable that is only in exponents?

Hi there. I've got this equation: $$\left(\frac{3}{5}\right)^x+\left(\frac{4}{5}\right)^x=1$$ How can I find the $x$? Thanks for helping!
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1answer
32 views

proving roots of equation $x^2+x+4f=0$ are real.

Let $a,b,c$ be three positive real numbers such that $a+b+c=1$. Let $f=\min(a^3+a^2bc, b^3+ab^2c,c^3+abc^2)$. Prove that the roots of the equation $x^2+x+4f=0$ are real.
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0answers
24 views

consider if the number is negative or positive

We have number $\displaystyle a= - \sum_{1\le}\sum_{i<j}\sum_{<k\le n} a_ia_ja_k + \sum_{1\le}\sum_{i<j}\sum_{<k < p}\sum_{\le n} a_ia_ja_ka_p - \cdots \pm a_ia_ja_ka_p\cdots a_n $ ...
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2answers
11 views

Variable with an exponent variable

I'm actually dealing with an economics problem, but it seems like the math is always what messes me up. Ignoring what the variables mean, I'm trying to understand how to get from step 1 to step 2. $$ ...
2
votes
3answers
36 views

Problem related to remainder

A polynomial in $x$ leaves a remainder $2$ and $3$ when divided by $x-1$ and $x+1$. What is the remainder, when divided by $x^2-1$ ?
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2answers
31 views

Eigen values and Eigen vectors

Let A be a 4x4 matrix with real entries such that $ \ -1,1,2,-2 \ $ are its eigen values.If $B=A^4-5A^2+5I$ ,where $I$ denotes the 4x4 identity matrix ,then which of the following statements are ...
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1answer
21 views

Finding two unknowns from equation [on hold]

I have $$\frac{P(\frac{L}{2})^3}{6}+C_1(\frac{L}{2})=\frac{PL(\frac{L}{2})^2}{4}+C_4$$ And in my textbook the next step states that $$C_1=-\frac{3}{8}PL^2$$ $$C_4=-\frac{11}{48}PL^3$$ But they do ...
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2answers
18 views

Simplify ((L/2)^3)/6

I have the expression $\frac{({\frac{L}{2}})^3}{6}$ but do not know the steps to simplify. Can someone please explain the steps for simplifying this. Thanks
0
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2answers
19 views

simplifying 3 different equations with 3 different variables

I am stuck trying to get the values for x, y, and z. I keep moving variables around but I end up getting answers like x = x or z = z and I do not think that is what I want. It's really just algebra ...
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2answers
39 views

Minimum value of a function

For $x \in [0, 5]$, let $$f(x) = \sum_{i = 1}^{5}\frac{1}{|x - i|}.$$ Why is $$f(x) \geq 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} = f(0)?$$ This of course is true if one simply plots ...
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0answers
14 views

Descartes rule of signs

I'm trying to write an algorithm that gets a polynomial and gives how many roots does it have in the interval [0 $x_0$]. I'm supposed to do it by Descartes law, I know that by Descartes law you Know ...
3
votes
3answers
104 views

which of $\sqrt{5}+\sqrt{13}$ or $ \sqrt{34}$ is larger?

which of $\sqrt{5}+\sqrt{13}$ or $ \sqrt{34}$ is larger? I tried to define $f= \sqrt{x}+\sqrt{x+8}+\sqrt{x+29}$ and use cslculus but have failed. please helps. Is there any non calculus solution? ...
0
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1answer
18 views

Word problem — water pipes and pool

Three pipes 1,2 and 3 fill together a pool in 6 minutes. Pipe 2 alone fills the pool in 75% of the time that pipe 1 alone fills a pool. Pipe 3 alone fills a pool in 10 minutes longer than pipe 2 ...
0
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2answers
39 views

$-5| 2+4x | = -32(x+3/4)- | x | + 1$

This was my attempt: $$-5| 2+4x | = -32\left(x+\frac34\right)- | x | + 1\\ \implies|2+4x|=\frac{-32x-24- | x | + 1}{-5}\\ \implies2+4x=\pm \frac{-32x+-24- x + 1}{-5}\\ \implies4x=\pm \frac{-33x+-23 ...
3
votes
3answers
60 views

Nice parameterization of $x^2 + y^2 - kx^2y^2 =1$

Can anyone find a nice simple parameterization of this curve. Just the quarter where $x \ge0$ and $y \ge0$ would be fine. The parameterization should be "nice" in the sense that the first derivative ...
0
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1answer
16 views

Converting shares from the chemical disassociation equation into fractions

Some basic math is eluding me when trying to derive a simple disassociation constant formula. Given that $K_d=\frac{[A][B]}{[AB]}$, $[A]+[AB]=[A_0]$, $[B]+[AB]=[B_0]$, and $[B_0] \gg [A_0]$ I'm ...
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0answers
25 views

Convolving two functions

I'm trying to convolve two functions $f$ and $g$. $$f(x) = e^{-\frac{{(x-p_2)}^2}{2 q_2^2}}$$ $$g(x) = \left(i_1 e^{-\frac{(a-x)^2}{2 \sigma ^2}}+j_1 e^{-\frac{(b-x)^2}{2 \sigma ^2}}\right) \left(i_0 ...
0
votes
1answer
22 views

Polar graph question

Can you only graph periodic functions using polar graphing? I'm not really understanding this I guess. It you are to get all of the x and y values on a finite graph, then the original must be ...
0
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1answer
10 views

Linear programming problem answer

Big Seas makes regular ice cream and non fat ice cream. The ice cream mixer can make at most 300 gallons. Each regular ice cream requires 5 ounces of milk, and each non fat ice cream requires 2 ounces ...
0
votes
1answer
15 views

What expression represents the total cost?

A customer calculated the cost of a new jacket , c, including a 7% sales tax, by multiplying 0.07 times the cost of the jacket and adding the product to the cost of the jacket. What is another way to ...
0
votes
0answers
25 views

Distance between point A and and point B.

A surveyor on one side of a river wishes to find the distance between points A and B on the opposite side of the river. On her side, she chooses points C and D, which are CD = 20 m apart, and ...
0
votes
2answers
17 views

How many ounces of water is necessary to dillute an active ingredient in a solution?

Solution A has 10% active ingredient and 90% inactive ingredient. You have exactly 0.25 ounces of solution A. How many ounces of water must be added to solution A to dilute the active ingredient to ...
0
votes
2answers
67 views

How to solve this inequality?

I have the following problem in an assignment and have been struggling to do it. $2 + 2x - x^2 \geq 2 \sqrt{1+2x}$ I have tried solving for $x$ but have not been able to do so. Any hints to solve ...
0
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2answers
69 views

How many days are there in 70 years?

How to calculate the total no. of days in 70 years (or any other no. of years) considering that this period also includes leap years?
2
votes
1answer
58 views

$\sqrt[\large m]{(x+y)}\over \sqrt[\large k]{(x+y)}$ $=\sqrt[\large m-k]{(x+y)} $?

Is it always true that: $\sqrt[\large m]{(x+y)}\over \sqrt[\large k]{(x+y)}$ $=\sqrt[\large m-k]{(x+y)} $ where $m,k \in \mathbb N$ ? I tried it with a few numbers and it seems to work every time.
1
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0answers
18 views

How to approach sketching sine and cosine graphs with transformations

Any tips or suggestions in sketching these graphs quickly, and in ONE go? In exams, I don't want to spend ages re-drawing the original sine/cosine graph, one by one, following each new ...
2
votes
6answers
50 views

General solution for squared trigonometry questions: $\cos^2 x = 1$

$\cos^2 x = 1$ How do you solve trig equations with a power? Unsure what to do with the square? I get this $\frac{1+\cos2x}2 =1$ $\cos2x =1$ $2x=2n\pi\pm0$ $x=n\pi$ but the answer says $\pm ...
0
votes
1answer
37 views

How does a sequence's convergency change finite sums?

What has been troubling me lately is that I cannot grasp how a finite series could ever diverge if a finite sequence that is divergent can only imply to a finite sum every time. Perhaps my main ...
5
votes
0answers
45 views

How can I better solve proofs requiring the introduction of algebraic assumptions?

Today I decided to binge on discrete mathematics after a three year hiatus. I tackled three proofs, and all of them required the introduction of assumptions that seemed to not be found in the givens ...
1
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5answers
150 views

Algebra problem stumping me

I have recently run into an algebra problem that goes as follows. Using the digits $1$ to $9$, $$ \left\{ \begin{align} A + B + C + D &= EF \\ E + F + G + H &= CJ \\ B + G + J ...
0
votes
1answer
20 views

Maximizing the area of the rectangular part of a running track only

Question: A high school is planning to build a new playing field surrounded by a running track. The track coach wants two laps around the track to be 1000 m. The football coach wants the rectangular ...
0
votes
0answers
34 views

Simplifying a rather long expression

I'm struggling to simplify this expression: $$ i_1 j_0 \exp \left(\frac{-81 a^2-2 p_2 \left(65 a+49 b+57 p_2-114 x\right)+2 p_1 \left(16 a-65 b-49 p_2+49 x\right)+32 a b+130 a x-65 b^2+98 b x-65 ...
0
votes
3answers
49 views

Finding the limit of $\lim_{ x\rightarrow 3^-} ((\sqrt{3x+7}-4)/(\sqrt{3-x}))$

How do I evaluate $\lim\limits_{ x\rightarrow 3^-}\dfrac{\sqrt{3x+7}-4}{\sqrt{3-x}}$? Can someone explain the steps by steps solution to this problem?
0
votes
0answers
15 views

Reverse Sliding scales?

Is it possible to create a 'reverse' sliding scale? Here is what I mean: Value 1 Value 2 0...........200 ............. ............. ............. ............. ............. ...
1
vote
1answer
30 views

Finding the minimum value of a 6th degree polynomial algebraically

Is it possible to answer this question using methods of basic algebra? Find the least value of the expression $a^6 + a^4 - a^3 - a + 1$ for real value of $a$. This question is from the 2013 Philippine ...
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2answers
48 views

Finding roots of cubing equation

Find all roots of the following polynomial: $$x^3 + x^2 + 1$$
1
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3answers
41 views

Finding the limit of $\lim_{x\to 1} (x^2-\sqrt x)/(1-\sqrt x)$

How do I evaluate $$\lim_{x\to 1} \frac{(x^2-\sqrt x)}{(1-\sqrt x)}$$ Can someone explain the steps by steps solution to this problem?
4
votes
7answers
198 views

Which is bigger: $\sqrt{1001} - \sqrt{1000}$, or $\frac{1}{10}$?

Which is bigger: $\sqrt{1001} - \sqrt{1000}$, or $\frac{1}{10}$? I can calculate the answer using a calculator, however I suspect to do so may be missing the point of the question. The problem ...
2
votes
4answers
63 views

Computing coefficient of $x^n$

Find the coefficient of $x^n$ in the expansion of $$\left(1 + \frac{x}{1!} + \frac{x^2}{2!}+\cdots +\frac{x^n}{n!} \right)^2$$ How do you even start this problem? Do you use multinomial theorem or ...
2
votes
1answer
37 views

Formula for this pattern

I am trying to develop a computer program to compute the tax for a given base salary, I believe given the format of the income tax table that I have there should be a formula to calculate the tax for ...
5
votes
2answers
91 views

Solving complex trig functions: $\sin2x + \sin3x = \frac{\sqrt{3}}2$

How to solve: $$\sin(2x) + \sin(3x) = \frac{\sqrt{3}}{2}$$ where $x$ is in $[-\pi,\pi]$? I have no idea what to do with the $\sin(2x) + \sin(3x)$. Am I supposed to factorise, differentiate, is ...
1
vote
1answer
50 views

Why ${(a^2)}^{\frac 12}=\sqrt {a^2}=|a| \neq a$?

Let $a\in \mathbb R$. It should be true that $\sqrt {a^2}=|a|$, since $\sqrt {(-2)^2}=\sqrt{2^2}=2$ and so on. But, it is also true that ${(a^2)}^{\frac 12}=a$, and by definition, ${(a^2)}^{\frac ...
0
votes
1answer
16 views

$n$ is some natural number. Let $x$ be the integer part of $\sqrt n$ and $y$ be the decimal part. If $x^2 - y^2 = 1+4y$ what is $y^x$?

$n$ is some natural number. Let $x$ be the integer part of $\sqrt n$ and $y$ be the decimal part. If $x^2 - y^2 = 1+4y$ what is $y^x$? This is some high school problem but I can't solve it. Any help? ...
5
votes
2answers
88 views

A different type binomial expansion problem

Suppose we have $$(1+x+x^2)^n = a_0 + a_1 x + a_2 x^2 + \cdots + a_{2n} x^{2n}.$$ What will be the value of $a_0^2 - a_1^2 + a_2^2 - \cdots + a_{2n}^2$? The answer is $a_n$, but I can't solve it. ...
2
votes
1answer
26 views

finding vector that isn't a linear combination

Hi can someone help me with this question: Find a vector in $\mathbb{R}^5$ which is not a linear combination of u and v. Verify that your vector is not a linear combination of u and v. Where u = ...