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
18 views

Maintaining relative percentage increases with constraints on max and min values

Good morning, I'm building a scoring model and I need to have scores that go from a minimum of 50 to a maximum of 400. I have to come up with 15 scores. Their values don't matter as long as the ...
1
vote
4answers
122 views

Approximation to $\sqrt{\cos(\theta)}$?

I have this formula, (it is just the law of cosines angle formula): $$ d = \sqrt{a^2 + b^2 - 2ab \ cos(\theta)} $$ Here is my issue. I am wondering if there is a way to 'extract' the $cos$ term. My ...
0
votes
2answers
45 views

Problem in algebra VIII

So the problem is to solve the equation: $$ x^2-i=0$$ So I thought of factoring this out: $$ x^2-i=(x+\sqrt{i})(x-\sqrt{i})$$ Why? Because: ...
1
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1answer
76 views

Square root of x plus square root of x plus …

I want to confirm if my answer in this problem is correct: $$\sqrt{(x + \sqrt{(x + ...))}} = (1 + \sqrt{53}) / 2 $$ Solution: $$x + \sqrt{(x + \sqrt{(x + ...))}} = (1 + \sqrt{53})^2 / 4 $$ $$x + ...
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1answer
97 views

(Basic High School Mathematics) Graphing the inverse square law

I did an experiment measuring the intensity of light in relation to the distance away from a source. How would I graph the avg intensity over 1/distance squared? Note that T1 = trial 1 etc.. It's ...
-5
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1answer
74 views

Problem on Algebra [closed]

$$(a\alpha+b)^{-2}+(a\beta + b)^{-2} = 1,$$ where $\alpha$ and $\beta$ are the roots of $ax^2+bx+c=0$. Then $ac(ac+2)=$ A) $\frac{b^2}2$ B) $b^3$ C) $2b$ D) $b^2$
0
votes
4answers
55 views

if x^2 + 2x - 3 >= 0 then (x <= -3) V (x >= 1)

I know why this is true but putting it in symbolic notation has me stumped. so basically i have that: ...
0
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1answer
40 views

How to come up with formula for this number growth

I have difficulties to come up with a formula to achieve this in my programming challenge : The input number ranges between 100 to 10.000 (all integers), then the output would be 0.1 - 10, so the ...
0
votes
1answer
17 views

Compounding interest + continuing eposit

If i deposit $3,000$ dollars every year for $10$ years, with an annual compound of $10 \%$ return, what is the formula that I would use to figure this out? I can't wrap my head around this. I have ...
2
votes
7answers
129 views

for $n$ an integer, why is $n^0=1$ ??

This is so going to cost me.... I was wondering why for any integer $n$: $n^0 =1$. Perhaps It's because $n$ is a round number and if $m$ is a non negative integer, also round then: $$n^m = 1 \cdot ...
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3answers
33 views

Finding Vertical and Horizontal Asymptote lines

what is the vertical and horizontal asymptote lines of the equation $y=\frac{2}{x+1}+2$. I think the vertical asymptote line is $x=-1$ but am lost on how to find the horizontal asymptote line.
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1answer
33 views

Vertical and Horizontal Asymptote Lines

What is the vertical asymptote line and horizontal asymptote line of $y=1-\frac{1}{x}$? I have gotten the vertical asymptote line to be $x=0$ and the horizontal asymptote line to be $y=o$ but when I ...
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votes
2answers
38 views

How to find the number of atoms of an element in $x$ amount of a compound

Lets say I am given a certain mass of a compound. Additionally, the number of moles of the compound itself are known, as well as its molar mass. How can I, using an equation, find the number of atoms ...
0
votes
2answers
36 views

Prove that this equation have an non finite number of prime solutions

So the question seeks to answer the following, let $x,y\in\Bbb R$. Prove that there is a non finite number of prime solutions to the following equation: $3x-5y=11$. Our professor says that it's easy ...
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3answers
110 views

Find the range of values of $x$ for which $1-x<(x-1)(5-x)<3$.

Find the range of values of $x$ for which $1-x<(x-1)(5-x)<3$. First of all, I solved $1-x<(x-1)(5-x)<3$ which gives me $(x-1)(x-6)<0$ and $(x-4)(x-2)<0$. How to find the range, ...
0
votes
1answer
84 views

Finding the remainder of a polynomial P when divided by $x^2 -1$

I need help please to answer this problem: The remainder of a polynomial P (in one variable $x$) when divided by $x^2 -1$ is a polynomial of degree at most 1, that is, it has the form $ax + b$ ...
2
votes
2answers
174 views

Finding the “triangular root” of a number.

A triangular number is a number that is the sum of the natural numbers up to some $n$. The closed form is $x = \frac{n(n+1)}{2}$. How do I get $n$ on one side? I've been looking at it from every ...
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4answers
82 views

Pre College Mathematics

During my school days I was a very keen student of mathematics. But circumstances led me to opt for commerce at the college level. Now I wish to continue learning mathematics on a self study basis. ...
0
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1answer
27 views

Removing constants out of square roots

I have the equation $$ z = \dfrac{ħ}{\sqrt{2m_e T k_B \ln 2}} $$ Where $2, \hbar, m_e , k_B , \ln(2) $ are constants. What I want to do is turn this equation into the form $$ z = k_1 * ...
1
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2answers
41 views

Find a recurrence for in , the number of integer compositions of n which only have 1s and 2s as parts.

Find a recurrence for $$i_n$$ the number of integer compositions of $n$ which only have $1$s and $2$s as parts. How do you approach this problem?
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1answer
34 views

Rearranging the terms so that the denominator becomes the numerator

I have the equation $$ \frac{120}{1 + 3.167 \cdot e^{-0.05t}} = 60 $$ How do I transform it so that the denominator becomes the numerator? This would make the problem much easier.
1
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1answer
107 views

Find rectangular equation of a cardioid

Given the equation in polar form $$r = 1 - \sin\theta,$$ find the rectangular equation. So far, I found: $$x^2 + y^2 = 1 - 2\sin\theta + \sin^2\theta\quad x = \cos\theta - \sin\theta\cos\theta\quad ...
0
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1answer
67 views

Estimating powers like $1,000,000^{0.000001}$

I found a fun way to estimate powers like the above and am interested if there are even quicker ways. In the example given in the question start with $e^{(\ln 1,000,000)(0.000001)}$. Recognize ...
1
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3answers
32 views

using arctan to find the angle of a sloped line

When finding the angle of a sloped line using $\arctan(\frac{y}{x})$, when do you need to add $\pi$ to find the angle counterclockwise from the positive $x$-axis?
2
votes
1answer
52 views

Maximum of Sums of Product Pairs

Given two ascending distinct integer sets, $A = A(0), A(1), \dots, A(n)$, and $B = B(0), B(1), \dots, B(n)$, I'm looking for the maximum sum, where elements from $B$ are multiplied by elements from ...
1
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2answers
50 views

Is a function still a function if it doesn't have any rule?

From what I've read on the internet, I've concluded that function differs from relation in that function can only have one range per domain. So, if for example: ...
1
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4answers
135 views

Trigonometry inside a trapezium

I have the following image, and it's asked to find the values of $X$ and $Y$. I've managed to find it using the this idea: Divide the image in two right triangles and let's call the height of the ...
1
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1answer
62 views

A trigonometric equation with many cases

I have to solve the equation $4\cos^m(x)+3\sin^n(x)=5 $ where $m$ and $n$ are non-negative integers. So here comes my question: The case $m=n=0$ is trivial. The case $m=n=1$ is easy to solve. We ...
1
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1answer
54 views

Found an example for solving via quadratic formula in a book where I am wondering if this is correct

As a refresher, I was skimming through a free Calculus online textbook "MOOCULUS massive open online calculus" (https://mooculus.osu.edu/handouts) and stumbled upon the following example solving a ...
2
votes
1answer
67 views

How to solve $6^{2x}-10\cdot 6^x=-21$ using logarithms?

What do I do with $\large 6^{2x}-10\cdot 6^x=-21$? Since $6$ and $-60$ are not of the same base (nor can they be written as exponents of the same base cleanly) I am having trouble solving for ...
0
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2answers
89 views

Find Vector and Parametric Equation

I'm having some trouble finding answers to these problems. When i try to find help online, all i find are (x,y,z) problems and I'm simply looking for a PreCalculus (x,y) problem solving technique: ...
3
votes
2answers
61 views

How to prove infinitely many integer values for a square root equation?

I have the equation $y = \sqrt{3x^2 + 1}$, and I need to prove that there will be infinitely many integer solutions. I saw possible solutions with things like Pell's equation, but I did not fully ...
2
votes
0answers
42 views

How to solve the quadratic equation problem with a strict proof

Consider $f(x)=x^2-(a+b)x+ab$ with $n\le a\le b\le n+1$ where $n$ is a positive integer. Find the range of $\min\{f(n),f(n+1)\}$ Sorry I just made a mistake,now is fixed. This problem is obvious in ...
1
vote
1answer
42 views

How many points determine a two variable polynomial of degree n+k?

I am working with sequences and it would be extraordinarily useful to have a two variable version of the following: A degree-$n$ polynomial is uniquely characterized by its values at any $n+1$ ...
-2
votes
2answers
42 views

Find the range of values of $x$ for the inequality $x^2-4x-1>0$ [closed]

Find the range of values of $x$ for the inequality given. $x^2-4x-1>0$
0
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0answers
27 views

Does a Positive integer exist that can convert…

Can you prove or disprove the existence of a positive integer P, such that P can convert the expression; $6ab+a+b$ into the form $6xy+x-y$ by subtracting from it. What I am trying to find is a ...
2
votes
1answer
37 views

How to solve $y=-x^3/(x^2-9)$ for $x$

This is not a homework question. I am doing a independent study refresher on precalc prior to taking calculus. Wolfram gives an unbelievably long series of steps with techniques I have not even heard ...
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votes
2answers
46 views

How much did the restaurant lose? [closed]

A person eats a £70 meal at a restaurant and pays with a £100 pound check. the restaurant owner has no change, so goesto the shop next door and changes the check for cash, then pays the customer £30 ...
1
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1answer
70 views

Any book on mathematical demonstration

I've found many books which treats of the theoretical and conceptual part of the mathematical demonstration, but now I'm searching for a book with a plenty of demonstration of mathematical proofs. ...
2
votes
1answer
56 views

Express in terms of $x$ and $y$ when the values of $x$ and $y$ are given.

Given, $x=1+3a+6a^2+10a^3+\ldots$ $y=1+4b+10b^2+20b^3+\ldots$ $s=1+3ab+5(ab)^2+7(ab)^3+\ldots$ Express $s$ in terms of $x$ and $y$. My work: I could see how the first sequence works, but could not ...
2
votes
4answers
93 views

Simplfying $\sqrt{31-8\sqrt{15}}+\sqrt{31+8\sqrt{15}}$

I am trying to simplify the expression: $\sqrt{31-8\sqrt{15}}+\sqrt{31+8\sqrt{15}}$ I tried to square the expression but I can't do that because it is not an equation so I got stuck. Can someone ...
1
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1answer
27 views

Solving simultaneous equations in terms of variables

If $x+y = m$ and $x-y=n$ then $(x^2-y^2) -2x$ is equal to in terms of $m$ and $n$ only! How do you solve?
2
votes
1answer
36 views

How to conclude that $r^t(E\cos(\theta t)+Fsin(\theta t))$ ossilates with increasing magnitude?

Given: $r^t(E\cos(\theta t)+Fsin(\theta t))$ Assume $r>1$, $E>0$, $t\ge0$ and $F$ is not known. How do we conclude that the given expression oscillates with increasing magnitude? My ...
0
votes
1answer
51 views

How to tell if the roots to a quadratic equation is always positive using the quadratic formula?

Suppose $k>0$ and $(k+1)^2>8k$. Let: $\alpha,\beta = \frac{(k+1)/2 \pm \sqrt{(k+1)^2/4-2k}}{2}$ The solution I have says that $\alpha$ and $\beta$ will always be positive. I don't know how ...
0
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1answer
14 views

Re-arrange expression to transformation form

$$\frac{6x-5}{3x+1}$$ How do you write this in the form $$\frac{b}{x+c} + a$$ I know how to find a (2) by asymptote theory, but I don't know how to re-arrange to find B.
3
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2answers
61 views

How do I get from $\frac{-x+1}{-x+2}$ to $1 + \frac{1}{x-2}$

wolframalpha tells me it's the same but I can not follow how to get from one to another. $$\frac{-x+1}{-x+2} = \frac{1-x}{2-x} = \>? \dots$$ I don't get any further, always end up where I ...
4
votes
1answer
62 views

issues with simple algebraic equations

$ab + a + b = 250$ $bc + b + c = 300$ $ac + a + c = 216$ then find $a + b + c = ?$ MY APPROACH: (i) * c , (ii) * a , (iii) * b then we get $abc + ac + bc = 250c$ $abc + ab + ac = 300a$ ...
0
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1answer
60 views

Trigonometric eq.

The equation $3\sin(x)+4\cos(x)=5$ is well-known. The equation $3\sin^m(x)+4\cos^n(x)=5$ where $m$ and $n$ are non-negative integers is much more interesting.. I would like to see a nice, elementary ...
0
votes
1answer
65 views

Finding the value of $p$

I need some help with this question: Consider an individual who possesses the Bernoulli utility function of $u(x)=\dfrac{x^{1-\gamma }}{1-\gamma }$ where $\gamma>0$, $\gamma \neq 1$. Who maintains ...
1
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3answers
84 views

Pre Calculus Expression

The questions is: $$\dfrac{3(x+2)^2(x-3)^2 - 2(x+2)^3(x-3)}{(x-3)^4}$$ My answer is: $$\dfrac{3(x+2)^2 + 6x^2-4}{(x-3)^2}$$ Am I right? If not, where have I failed?