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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1answer
10 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
22 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 ...
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1answer
20 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 ...
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1answer
8 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 ...
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1answer
13 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 ...
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0answers
15 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 ...
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2answers
16 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 ...
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2answers
63 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 ...
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2answers
67 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
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1answer
52 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.
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0answers
17 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 ...
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6answers
44 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 ...
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1answer
34 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
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0answers
37 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 ...
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5answers
125 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 ...
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1answer
19 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 ...
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0answers
33 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 ...
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3answers
47 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?
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0answers
13 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
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1answer
24 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
46 views

Finding roots of cubing equation

Find all roots of the following polynomial: $$x^3 + x^2 + 1$$
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3answers
37 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?
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7answers
188 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 ...
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4answers
60 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
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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 ...
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2answers
86 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 ...
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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
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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
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2answers
83 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
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1answer
24 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 = ...
1
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1answer
22 views

Would every half angle of an angle in each quadrant be in the previous quadrant?

For example, take (5pi)/4 which is in Q3, it's half angle is (5pi)/8 which is in Q2. Is this true for every angle?
0
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1answer
36 views

Verifying the cosine rule

Verify the following system of linear equations in cos A, cos B , and cosC. Triangle cannot be shown. Then use Cramer’s Rule to solve for cosC , and use the result to verify the Law of Cosines: ...
1
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1answer
24 views

How many gaussian integers have modulus 5?

My work so far: $$0+5i\quad 5+0i\\ 3+4i\quad 4+3i$$ so there are 4 such integers. But when I type it in Brilliant I get wrong!! Maybe for them 0 is not an integer so I type 2 and I get wrong! What the ...
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3answers
48 views

What is the best way to find the roots?

In Calculus, mainly when we compute areas we face equations like $$x^\frac{1}{2}=x^2$$ I know that I can take the square root of the both sides. Does anyone know another way to find the roots of the ...
0
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1answer
52 views

Hint finding exact value of half-angle when $\tan (\theta) = {3}$

Unlike others I've tried, I'm having a hard time with this half-angle exercise: If $tan(\theta)={3}$ and $\theta$ is in QIII, find $\tan\left(\frac{\theta}{2}\right)$ Here's what I know (or think I ...
2
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6answers
211 views

Algebra: What does “is defined for” mean?

In algebra what does: "Is defined for" mean? I have a question posted: $\sqrt{a+b}$ is defined for $-b \leq a$. The question posed is: Is this true... My question: WHAT DOES "Is Defined For" ...
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1answer
16 views

System of inequalities. Points of intersection?

$x^2+y^2<=81$ $y<x$ Is this correct? My answer: (-9sqrt(2)/2,-9sqrt(2)/2), (9sqrt(2)/2,9sqrt(2)/2)
2
votes
3answers
61 views

If $ f(x)=x^2-3x+1$ then $ f(x-2) = ?$

If $ f(x)=x^2-3x+1$ then $ f(x-2) = ?$ I'm not sure how to properly deal with this function and solve for $f(x-2)$.
1
vote
1answer
20 views

Check my solution to this trig inequality

Problem $1.88$ : Solve $$\cos x\lt \frac{\sqrt{3}}{2},\qquad x \in [0,2\pi]$$ I found the set of solutions to be $S=[0,2\pi]-\left[\dfrac{\pi}{6},\dfrac{11\pi}{6}\right]$ Is this correct? Thank you.
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4answers
36 views

Prove that $\sin(\frac{\pi}{3}+x)=\cos(\frac{\pi}{6}-x)$

How to prove that $\sin(\frac{\pi}{3}+x)=\cos(\frac{\pi}{6}-x)$ without using calculus just trigonometry?
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0answers
16 views

What would be a formula to this “ trains ” meet problem?

Guy riding a bicycle with speed 9km/h leaves City A. One hour and 15 min later, a bike rider is leaving City B traveling towards City A with speed of 21km/h. At what distance/km would these two ...
4
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1answer
76 views

A tough inequality problem with condition $a+b+c+abc=4$

If, $a+b+c+abc=4$, with $a,b,c$ being positive reals, then prove or disprove the following inequality: $$\frac{a}{\sqrt{b+c}}+\frac{b}{\sqrt{a+c}}+\frac{c}{\sqrt{a+b}}\geq\frac{a+b+c}{\sqrt2}$$ I ...
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2answers
23 views

How to measure monotonicity of a list of values

I need to compare monotonicity of lists of values. I have $S=(n_1,n_2,...n_n)$, I need a function $\mathrm f(S)$ to return the monotonicity of the S. $S_1=[1,2,4,4,8]$ $S_2=[8,4,4,2,1]$ ...
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9answers
3k views

Kid's homework: 4 equations 5 unknowns? Going crazy!

I'm new here, and I'm hoping someone can help out. My 10 year old son has been set a maths problem, which I can't solve. I've got a PhD in neuroscience and do a fair amount of matlab stuff (data ...
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5answers
85 views

How to teach newbie multiply of complex number

I want to teach a newbie the arithmetic law of complex numbers. the law of add is acceptable psychological. but multiply is not. for example, assume $$z = a+bi, w = c+di$$ He (She) may ask me: why ...
2
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1answer
55 views

Bombelli's wild thought of cubic equations

In many books, like Visual Complex Analysis. talk about the real original of complex number. the author begin with this equation: $$x^3=15x+4$$ Then the author use the formula ...
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1answer
18 views

Intersection of graphs, and no solution for trig functions.

All I know is the c=asin(x-b) I don't know how to check the values of b for 'no solutions,' in the case of trig functions. Can someone people provide an algebraic method to solve this.
0
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1answer
88 views

Proof for A majorizes B

$\alpha = [\alpha_i] \in\mathbb R^n$ and $\beta = [\beta_i]$ where $\beta_1 = \beta_2 = ......=\beta_n = \frac{1}{n}\sum\alpha_i$ How can i show that $\alpha$ majorizes $\beta$ I tried to get a ...
0
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1answer
14 views

Choosing values for octave/decade scale

I am developing a frequency response simulator for linear circuits which should be able to plot graphs of Voltage x Frequency with the latest varying linearly, in octaves or in decades. The only ...
0
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2answers
41 views

Can this system be solved algebraically?

$$6x+zy=57\\ (4+x)(6+y)=81\\ z(6+y)=81$$ Is there a way to solve it algebraically? I already know the answer, I just need to know if it can be solved.