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
247 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
vote
1answer
728 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
votes
1answer
72 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
57 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
60 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
52 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
610 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
71 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
70 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
77 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
204 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
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2answers
83 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
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0answers
53 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
192 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 $n+1$ ...
0
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0answers
32 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
54 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 ...
1
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1answer
120 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
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1answer
62 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
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4answers
105 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
vote
1answer
35 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
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1answer
39 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
86 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
104 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
63 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
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1answer
75 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
votes
1answer
61 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
84 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
94 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?
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4answers
61 views

Do not understand formula…

How does; $x^{n} -y^{n}=(x-y)(x^{n-1} + x^{n-2}y+...+x y^{n-2}+ y^{n-1} )$ work on $x^{2} - y^{2}$ When I attempt to apply the formula on $x^{2} - y^{2}$ I get the following $x^{2} - y^{2} ...
0
votes
1answer
176 views

Whats happens if you move a fraction into the denominator?

I have $(x^2 + 1) \frac{1}{3}(x^2 + 2)^{-2/3} (2x)$ and I see the solution is $ \frac{(1 + x^2) 2x}{3(2+x^2)^{2/3}}$. How come there is a $3$ on the bottom and not $1/3$? I thought a negative exponent ...
0
votes
2answers
65 views

Pre Calculus Question

The problem is a precalculus problem. $$\frac{\large\frac{1}{1+x+h} - \frac{1}{1+x}}{h}$$ I was wondering if I can use the distributive property by dividing out the denominators in the numerator. ...
1
vote
1answer
58 views

Why is $f'(x) $ of $y=\sqrt{5x} \neq 0$

Why is $f'(x) $ of $y=\sqrt{5x} \neq 0$? I would have worked it out like this: $f(x) =\sqrt 5.\sqrt{x} \equiv 5^{1/2}.x^{1/2}$ so $\dfrac{dy}{dx} = 0.(1)\left(\dfrac{1}{2}\right)x^{-1/2}$ since ...
0
votes
3answers
27 views

minimizing the value of a simple expression

I would like to minimize this very simple expression, and understand that the square root of 20 would offer a minimal value > 0: $$n + \frac{20}{n}$$ Can anyone explain or prove why $\sqrt{20}$ ...
0
votes
1answer
312 views

A Lower bounds for exponential function

How to find a "nonnegative" lower bounds for $e^{uX-u^2Y} + e^{-uX-u^2Y} - 2 $? where $u>0$, $Y \geq 0$ and $X \neq0$ I know I can bound above function by using the fact that $e^{uX} + e^{-uX} ...
0
votes
0answers
17 views

What parameter settings make this expression $\Omega(1)$?

I feel a bit silly for asking this, but here goes... I have the expression: $$(1 - (1 - n^{c+d-1})^{n^{2d}})^{n^{2-2d}}$$ We can assume $d$ is a constant somewhere in the range $[0, \frac{1}{3}]$. ...
1
vote
1answer
108 views

Does an expression exist such that…

Can you prove or disprove the existence of an expression P, such that $Z=6ab+a+b-P$ Makes Z expressible in the form; $Z=6xy\pm x \pm y$ for all a and b where $a,b,x,y∈N $ Finding an example of ...
0
votes
2answers
32 views

Annual Conversion: How-T0

I'd like to know how to convert prices -- $\$20$ per month is how much in one year? For example: "Tim pays $\$20$ a month for his cell phone bill. How much will he pay in one year?"
1
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0answers
67 views

Proof by induction for all positive integers

$\sum_{j=1}^n j^2 = \frac{n(n+1)(2n + 1)}{6}$ So I did the obvious and plugged one to show that $1^2 = \frac{1(1+1)(2(1) + 1)}{6}$ Now I am trying to show that $1^2 + 2^2 + ... + n^2 + (n + 1)^2 = ...
0
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2answers
57 views

Should one square both sides of the equation $\sqrt{1+2\sin{2x}}= \cos{x}-\sin{x}$ in order to solve it, or is there a better way?

We have $\sqrt{1+2\sin{2x}}= \cos{x}-\sin{x}$ Fistly my conception was squaring both sides but I figured out that this method is wrong, so my question is how should look solution this example ?
9
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3answers
2k views

Is it possible to solve this equation by hand?

I am working on a physics task, and reduced it to the following equation for $y$: $$\frac{1}{4y^3}-\frac{2}{(y^2+b^2)^{\frac{3}{2}}}=0$$ I handed it to Mathematica, and it gave me two real solutions, ...
1
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4answers
1k views

Find the range of values of $x$ which satisfies the inequality.

Find the range of values of $x$ which satisfies the inequality $(2x+1)(3x-1)<14$. I have done more similar sums and I know how to solve it. I tried this one too but my answer doesn't matches the ...
0
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2answers
38 views

Adding like terms for factoring

I am trying to factor the following identity: $4a^3 + 8a^2b^2 - 4ab^2 - 2a^2b$ When I first look at this, I say to myself I obviously need to simplify it further, since I can spot a few like terms. ...
1
vote
3answers
67 views

Formula for finding $a$ by $b$ and $c$

I have the formula: $$ c = \frac{a}{100(a + b)} $$ How to find $a$ by $b$ and $c$? $$ a = \text{?} $$
0
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2answers
60 views

Equation solving, why does this simplification work?

$$\dfrac{154}{3.2-x} = \dfrac{66}{1.6-x}$$ How can that equation be simplified as such? : $$154(1.6-x) = 66(3.2-x)$$ Can someone explain?
0
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3answers
114 views

This is regarding a verbal math problem

A Shopkeeper buys sweets at 11 for a rupee. He bought an equal number of sweets at 10 for a rupee. he sold all the sweets at 8 for a rupee. find loss or gain % Please solve this for me. I am not able ...
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3answers
43 views

How to move from powers to simple logarithms?

I'm following a book that briefly moves from $$16000 \times 2^{\displaystyle \left (-\frac{x}{24} \right )} = 1600$$ to $$x = \frac{24 (\log(2) + \log(5))}{\log(2)}$$ adding the comments that ...
0
votes
1answer
57 views

How to simplify floor polynomial given lower bound on x?

$$ \left\lfloor\frac{8x^2 + 5x -4}{3x^2 + x}\right\rfloor $$ where $x$ > $\sqrt{8}$ How would you simplify this type of expression? *Please note the floor operation surrounding the expression ...
0
votes
3answers
49 views

When is this statement true?

Under what conditions is the following equation true $$x_1=x_2+x_2x_1^2-x_1x_2^2 $$ I thought it was true only when $x_1=x_2$, but apparently there are more possibilities than this. Any help would ...
0
votes
4answers
58 views

For all $x$ in the set of reals, $ |x-2| > 2 \implies x^2 > 4x$.

How do i go on proving this statement? The first step i took was to assume the antecedent. so.. assume $|x-2| > 2$ then $x > 4$ or $x < 0$ if we assume the consequence, $x^2 > 4x$ ...
0
votes
3answers
69 views

How to generically solve polynomial expressions given a minimum value of x?

Given something like $\dfrac{8x^2 + 2x + 7}{3x^2 + 2x}$, and $x > \sqrt8$, what strategy would you employ to simplify this expression?