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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1
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1answer
21 views

Solving two equations with 2 variables

I am wondering if this equations can be solved by "a" and "b": b = 1 + 0.31*a a = c1 - c2/b c1 and c2 are constants, but change depending on some initial assumptions. One example of their ...
0
votes
2answers
23 views

Simplifying Cube Roots Containing a Square Root

I was doing a problem today, and arrived at the (correct) answer of $x^3 = 16000\sqrt2$ Obviously I want to simplify this further. My text book jumps straight to $x = 20\sqrt2$ with no explanation. ...
3
votes
3answers
44 views

Give the equations that are a tangent to the parabola $y = x^2 + 5x + 6$ and pass through $(1,1)$

I have been given the question: Give the equations that are a tangent to the parabola: $y = x^2 + 5x + 6$ and pass through the point $(1,1)$ I have tried two different methods for solving this. ...
1
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3answers
33 views

Is there an integer solution to $x^2+1978=y^2$

Is there an integer solution to $x^2+1978=y^2$? Don't know really how to approach this. Thanks
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5answers
53 views

Are these two expression equal?

My friend insisted that $(-1)^{(-n)}$ is equivalent to $(-1)^n$ for any number of $n$. A quick check in the Wolfram Alpha show ...
3
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4answers
51 views

Trying to solve the trig equation $\sqrt{3+4\cos^2(x)}=\frac{\sin(x)}{\sqrt 3}+3\cos(x)$

The equation is $$\sqrt{3+4\cos^2(x)}=\frac{\sin(x)}{\sqrt 3}+3\cos(x)$$ My solution goes like this $$ \begin{cases} 3+4\cos^2(x)=\frac{\sin^2(x)}{3}+\frac{6}{\sqrt 3}\sin(x)\cos(x)+9\cos^2(x) \\ ...
0
votes
2answers
21 views

Getting two different sets of results for $\sqrt{17+7\sin(2x)}=3\sin(x)+5\cos(x)$

The equation is $$\sqrt{17+7\sin(2x)}=3\sin(x)+5\cos(x)$$ My solution is, first, to define a system: $$ \begin{cases} 17+7\sin(2x)=(3\sin(x)+5\cos(x))^2 \\ 3\sin(x)+5\cos(x)\ge 0 \end{cases} $$ ...
0
votes
2answers
29 views

What is wrong with this formula?

I'm trying to make a formula that converts an ellipse in general form to one in standard. My steps to derive it are as follows: $$ax^2+bx+cy^2+dx+e=0$$ Move e to the other side... ...
2
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0answers
27 views

Let $A_1 A_2 \dotsb A_{11}$ be a regular 11-gon inscribed in a circle of radius 2.

Let $A_1 A_2 \dotsb A_{11}$ be a regular 11-gon inscribed in a circle of radius 2. Let $P$ be a point, such that the distance from $P$ to the center of the circle is 3. Find $PA_1^2 + PA_2^2 + \dots ...
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0answers
36 views

Equilateral triangle [on hold]

An equilateral triangle is one in which all three sides are of equal length. If two vertices of an equilateral triangle are $(0,\,4)$ and $(0,\,0)$, find the third vertex. How many triangles are ...
0
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0answers
39 views

Solving three quadratic simultaneous equations with three variables

I need to solve the following simultaneous equations: $$(2-a)^2+(3-b)^2+(-5-c)^2=6$$ $$(1-a)^2+(2-b)^2+(-3-c)^2=6$$ $$a+b+c=0$$ I've tried expanding and doing it the long way, but I don't ...
2
votes
3answers
90 views

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 ...
2
votes
1answer
51 views

How to show $\binom{2n}{n} \ge \prod_{n < p \le 2n} p $?

What is the best way to show \begin{equation} \binom{2n}{n} \ge \prod_{n < p \le 2n} p \end{equation} for prime $p$. I know that $ 2^{2n} = (1+1)^{2n} \ge \binom{2n}{n}$. and \begin{equation} ...
2
votes
1answer
48 views

Difficult sets of Equations, counting

Let $ m$ be the number of solutions in positive integers to the equation $ 4x+3y+2z=2009$, and let $ n$ be the number of solutions in positive integers to the equation $ 4x+3y+2z=2000$. Find the ...
3
votes
1answer
45 views

Find the sum of the roots of the floor equation

How to find the sum of the roots of the following floor equation? $$[\frac{x}{2}]+[\frac{x}{3}]+[\frac{x}{5}]=x$$ I found the following solutions by Mathematica: $\{\{ x= 0\},\{x = 6\},\{x = ...
0
votes
0answers
48 views

Find the value of $\frac {a+b+c}{x+y+z}$

$a^2+b^2+c^2=15\space \space$ $x^2+y^2+z^2=25$ $ax+by+cz=10$ Find the value of $\frac {a+b+c}{x+y+z}$ Thanks for any help.
0
votes
0answers
54 views

why $\frac{a}{b}\pmod p=\frac{a\pmod p}{b\pmod p}$

It is said this following is theorem? what's this name? and How to prove it? Thanks show that $$\dfrac{a}{b}\pmod p=\dfrac{a\pmod p}{b\pmod p},a,b\in N^{+},(a,p)=1,(b,p)=1$$
0
votes
2answers
42 views

Linear Equation in 4 variables- No of solutions

If 3a+6b+9c+4d = 100 and a ,b,c and d are natural numbers , then how many values d can take? How to approach this type of problem?
0
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0answers
35 views

What method(s) can be employed to solve this equation?

Solve for $n$ in the following equation: $$ 0 = 1780*1.006^n - 37n $$ Here's what I've tried: $$\begin{align*} 0 &= 1780*1.006^n - 37n \\ 37n &= 1780*1.006^n \\ \frac{37n}{1780} &= 1.006^n ...
14
votes
3answers
2k views

How to solve equations to the fourth power?

Is it possible to manually retrieve the value of $y$ from the following equation $$153y^2-y^4=1296$$ WolframAlpha has four solutions for $y$: $-12, -3, 3, 12$. How has it solved? What I've achieved ...
1
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4answers
57 views

Trying to solve $\sqrt{2\cos^2(x)-\sqrt{3}}+\sqrt2 \sin(x)=0$

The equation is $$\sqrt{2\cos^2(x)-\sqrt{3}}+\sqrt2 \sin(x)=0$$ I solve it thus: $$ \begin{cases} 2\cos^2(x)-\sqrt3=2\sin^2(x) \\ -\sqrt2 \sin(x)\ge 0 \iff \sin(x)\le 0 \end{cases} $$ The first ...
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0answers
16 views

prove the given question [on hold]

Prove that $\sec(2 \alpha)\cos(45^{\circ}-\alpha)\sin(45^{\circ}+\alpha) = \dfrac{1}{2}$.
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votes
1answer
30 views

Choose a variable to represent the number in parentheses.. [on hold]

The distance traveled in 3 h of driving was 210 km. ( hourly rate).... also write an equation that represents the given information
-3
votes
0answers
34 views

Speed/Distance math problem [on hold]

A corrections Canada transportation vehicle needs to travel $660$km. The vehicle travels at $100$km an hour for two hours. The driver stopped for gas for approximately $30$ minutes and when resumed ...
1
vote
0answers
14 views

Writing a word problem as a function.

I would like to verify that this word problem was translated into a function correctly. A towing company charges a flat rate of $100.97$ dollars per day plus $0.81$ dollars per mile. The ...
0
votes
0answers
17 views

Comparing Coefficients

If I have the equation: $4m(m-1)x^m .\sum_{i\geq 0}a_ix^i+x^m.\sum_{i\geq 0}a_ix^i=0$ ; $a_0\neq 0$ why am I able to say that $4m(m-1)+1=0$? I would understand if the equation rather than being an ...
1
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6answers
119 views

Prove $((a+b)/2)^n\leq (a^n+b^n)/2$

Struggling with this proof. Prove that $$\left(\frac{a+b}{2}\right)^n≤\frac{a^n+b^n}{2},$$ where $a$ and $b$ are real numbers such that $a+b≥0$ and $n$ is a positive integer. What technique would ...
3
votes
2answers
46 views

Polynomial with real roots

Consider the polynomial: $$f=X^4+4X^3+6X^2+aX+b$$ We know that $f$ has four real roots. Let $x_1,x_2,x_3,x_4$ be the roots of this polynomial. How can one compute ...
0
votes
2answers
76 views

How many pairs $(m, n)$ exist?

For certain pairs $ (m,n)$ of positive integers with $ m\ge n$ there are exactly $ 50$ distinct positive integers $ k$ such that $ |\log m - \log k| < \log n$. Find the sum of all possible ...
-1
votes
3answers
73 views

How do i solve $(2+i)z^2-(5-i)z+2-2i=0$? [on hold]

I need help solving $$(2+i)z^2-(5-i)z+2-2i=0$$ I have no idea how to start solving this equation so help would be nice.
3
votes
1answer
60 views

Chance of Drawing All of a Subset

I have a simple question but I can't seem to find the answer anywhere. Say that I have a set $\mathbb Z$ and a subset of that $\mathbb X$. I want to draw elements from $\mathbb Z$ until there is at ...
2
votes
0answers
52 views

Will $x=0$ satisfy the equation $\sqrt{\tan(3x)}=\sqrt{-\tan(x)}$?

The equation is $$\sqrt{\tan(3x)}=\sqrt{-\tan(x)}$$ And the one condition set for the solution is that $x$ should fall within this range: $0\le x < \pi$ The solution process boils down to $$ ...
1
vote
1answer
80 views

Solve $\log_9 (a) + \log_{12} (b) = \log_{16} (a+b)$ for $a/b$

The question: $$\log_9 (a) + \log_{12} (b) = \log_{16} (a+b)$$ solve for $a/b$. It gives hints: put it all in terms of x. $$9^x=a$$ $$12^x=b$$ $$16^x=a+b$$ Now prove that: $b^2=a(a+b)$ I did and ...
0
votes
1answer
58 views

Find the minimum value of $P=\frac{1}{2-x}+\frac{1}{2-y}+\frac{1}{2-z}$

Let $x,y,z$ be positive real numbers such that $x^3+y^3+z^3=3$. Find the minimum value of $$P=\frac{1}{2-x}+\frac{1}{2-y}+\frac{1}{2-z}.$$ I think that we need to show that $\dfrac{1}{2-x} \ge ...
0
votes
1answer
26 views

Solving intersection points

How do you solve these functions for intersection points? $2^x = 3-x$ Do you use natural log first to get the x or do you need to use other approaches?
0
votes
0answers
26 views

Rate of change problem. Over which interval is the function increasing? Over which intervals the function decreasing?

I would just like to verify that this solution is correct. Thanks. The table shows the number of animals killed by planes in West Africa from 1986 through 1990. Suppose the number of animals ...
14
votes
4answers
1k views

Every year, there is a contest…

Every year, there is a contest to see who has the heaviest pumpkins for that year. Last year, a farmer brought 5 pumpkins to the contest. Instead of weighing them one at a time, he informed the ...
0
votes
1answer
35 views

How to create quadratic equation given $y$ intercept, and maximum and $B=8$?

The given are Two x-intercepts y-intercept(0,-4) Maximum at (2,4) i tried everything i know...its been a long time since I have been doing math problems but the only way i thought about was to use ...
2
votes
1answer
17 views

Solving For Given Variable - How to Solve

So the question states: 'To convert degrees Celsius to Kelvin, the formula $K = C + 273.15$ is used. Solve this formula for $C$.' The answer I came up with was: $K - 273.15 = C$. So is this correct? ...
2
votes
2answers
27 views

Simplifying a complication max operation

I have dervied an inequality and have arrived to the following $$\max\{1, \frac{b}{2}+1\} \leq \max\{a, \frac{b}{2}+ \frac{a}{2}\}$$ I am trying to simplify further and arrived to the following ...
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votes
1answer
30 views

about simple interest

John invested certain sum in the three different schemes $P$,$Q$ and $R$ with rates of interest $10$% per annum and $12$% per annum and $15$% per annum respectively. If total interest accrued in 1 ...
1
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4answers
48 views

Confused about rules for solving systems of linear equations

Why are we allowed to add equations together/eliminate variables when solving systems of linear equations? I get that it works to find the solution but I don't understand why it works. Also, why ...
3
votes
1answer
26 views

How to solve (0.1 - 10.3 + 5.132)/12.8 and round off correctly?

I've recently learnt the rules about rounding off when adding/subtracting and when multiplying/dividing. I know that when you add/subtract, the number of decimal places in the result should equal the ...
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votes
1answer
30 views

Distribute coins fairly

$10$ coins weigh $10$ grams each, and another $10$ coins weigh $11$ grams each. So the average weight is: $$\frac{10 \cdot 10 + 10 \cdot 11}{20} = 10.5\text{ grams}.$$ Now I need to distribute coins ...
3
votes
3answers
35 views

An upper bound for a function

I am trying to find an upper bound $b\ge f(x)~\forall x\ge0$ for the following function: $$f(x)=\frac{x}{(w+ux^2)^2},$$ where $w,u>0$ are parameter values. I am interested in the positive domain ...
1
vote
4answers
46 views

The set of real values of $x$ satisfying the equation $\left[\frac{3}{x}\right]+\left[\frac{4}{x}\right]=5$

The set of real values of $x$ satisfying the equation $\left[\frac{3}{x}\right]+\left[\frac{4}{x}\right]=5$,(where $[]$ denotes the greatest integer function) belongs to the interval ...
4
votes
6answers
110 views

How to find $ab+cd$ given that $a^2+b^2=c^2+d^2=1$ and $ac+bd=0$?

It is given that $a^2+b^2=c^2+d^2=1 $ And it is also given that $ac+bd=0$ What then is the value of $ab+cd$ ?
2
votes
3answers
102 views

Probability that the eventually a six on a dice will appear.

Dave rolls a fair six-sided die until a six appears for the first time. Independently, Linda rolls a fair six-sided die until a six appears for the first time. Let $ m$ and $ n$ be relatively prime ...
1
vote
1answer
53 views

a simple question: whence the $\pi$ symbols in the solution of a trig equation?

There's a step-by-step discussion of an example irrational trig equation in my textbook. $$\sqrt{3\sin(2x)}=\sqrt{-5\cos(x)\cot(x)}$$ One of the solutions is $$\cos(x)=-\frac23$$ The solution to ...
4
votes
1answer
69 views

When is $(a+b)^n \equiv a^n+b^n$?

I remember a relation like $(a+b)^n \equiv a^n+b^n$, but I don't remember mod which numbers this is true. Where can I learn more about this?