Questions tagged [algebra-precalculus]

For questions about algebra and precalculus topics, which include linear, exponential, logarithmic, polynomial, rational, and trigonometric functions; conic sections, binomial, surds, graphs and transformations of graphs, solving equations and systems of equations; and other symbolic manipulation topics.

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1 vote
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20 views

Translating regular wording into a mathematical expression

Given a product that initially costs $2000$ dollars and will cost $200$ dollars more on each subsequent purchase (for example a total of $6600$ spent on the third purchase), how would I write a ...
7 votes
2 answers
81 views

Solving for x in logarithmic equation $\log_4(2x) = \frac{1}{2}x^2 - 1$

I am trying to solve for $x$ in the equation $\log_4(2x) = \frac{1}{2}x^2 - 1$. I have tried converting the logarithmic expression to exponential form, but I am not able to isolate $x$ in the ...
-2 votes
1 answer
36 views

Question about Exponents Rule

On what ground can $1/b$ = $a^{(3b-1)}$ be identified as $b = a^{(1-3b)}$. . Thank you EDIT: To be fully understood, my question is regarding identity. Here is my way, I'm not sure if the following ...
6 votes
1 answer
51 views

Solve $\frac{x-8}{x-10}+\frac{x-4}{x-6}=\frac{x-5}{x-7}+\frac{x-7}{x-9}$

Solve $\frac{x-8}{x-10}+\frac{x-4}{x-6}=\frac{x-5}{x-7}+\frac{x-7}{x-9}$ $\Rightarrow \frac{(x-10)+2}{x-10}+\frac{(x-6)+2}{x-6}=\frac{(x-7)+2}{x-7}+\frac{(x-9)+2}{x-9} \ \ \ ...(1)$ $\Rightarrow 1+\...
-3 votes
0 answers
27 views

Find the angles of intersection. [closed]

Find the angles of intersection between the curves $f(x) = x^2$ and $g(x) = \sqrt x$.
-3 votes
0 answers
20 views

Trying to find solution to a problem that involves undefined or ∞ values [closed]

I am trying to solve a problem which states "What is the sum of all m for which the value of 2/[m + 3/(m+4) ]is undefined?" These are the steps Step 1: 2/[m + 3/(m+4) ]is undefined when [m ...
-3 votes
4 answers
84 views

Why the Limit of $\sqrt x$ is infinity? [closed]

I'm having problems understanding why $\lim_{x\to \infty} \sqrt{x} = +\infty$. Can anyone please explain me why the result is positive infinity.
3 votes
2 answers
83 views

Simplifying the surd $\sqrt{3 - \sqrt{8}}$

Recently I was solving a math problem and I came across the following ( Only part of the problem ): $$\sqrt{3 - \sqrt{8}}$$ Here's is what I did to simplify the above: $$(a-b)^2=a^2+b^2-2ab$$ $$\sqrt{...
-4 votes
0 answers
21 views

f(x) is neither differentiable nor continuous anywhere except at x=0 [closed]

بِسْمِ اللَّهِ الرَّحْمَٰنِ الرَّحِيمِ f(x)=x^2 x∈Q f(x)=-x^2 x∈I I wish to show that f(x) is neither continuous nor differentiable except at x=0.
0 votes
0 answers
38 views

Is there a concise way to solve $T = S + ( ( S - E ) * N )$ without needing to enter $S$ twice?

If I have a starting number S and an ending number E, and it can be repeated N times, what will be the final total T? For example: S = 3540 E = 3650 N = 8 I can ...
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-7 votes
0 answers
31 views

What’s the answer to this problem? [closed]

$$\sin^2 a - \cos^2 a = 1 - 2 \cos^2 a$$
-2 votes
0 answers
19 views

Given a function, $h(x) = f(g(x))$, is there a way to express $g(x)$ in terms of $h$ and $f$? [closed]

Given a function, $h(x) = f(g(x))$. Is there a way to express $g(x)$ in terms of $h$ and $f$ ?
0 votes
1 answer
52 views

The Reflective Property of Parabolas [closed]

I am seeking help with the following math problem and would appreciate any assistance. The problem is as follows: A vertical beam hits a parabolic function $f$ at an arbitrary point $P(c,f(c))$. The ...
0 votes
0 answers
14 views

Summing a sequence of orthogonal complex exponentials

I have observed a very interesting behaviour which I would like to explain analytically, Specifically, I have observed $$\left |\frac{1}{N}\sum_{n=1}^{N}\exp(j2\pi f_nt)\,\Pi(t)\right |\approx\left |\...
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2 votes
2 answers
79 views

Quadratic polynomials $f(x)$ with rational coefficients such that $f(p)=q, f(q)=r, f(r)=p$ where $p,q,r$ are the roots of $x^3-7x+7$.

Let $p,q,r$ be the three roots of $x^3-7x+7$. Find all quadratic polynomials $f(x)$ with rational coefficients such that $f(p)=q, f(q)=r, f(r)=p$. So far, I let $f(x)=ax^2+bx+c$, and I have $$ap^2+bp+...
0 votes
0 answers
49 views

number of integer solution of quadratic equation.

Number of integral solution of $(k+1)x^2-2kx+(k+1)=0$ What I Try :: Let $\alpha,\beta$ be the integer roots of given equation. Then we have $\displaystyle \alpha+\beta=\frac{2k}{k+1}$ and $\...
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1 vote
0 answers
63 views

Generalization of the sliding ladder problem

Suppose we have a ladder of length $1$, and it's sliding down the $y-$axis. We know that the curve enveloped by it is an astroid: However, what if we iterate this process? We call this astroid curve $...
-3 votes
0 answers
17 views

Simplify formula with log function as exponent [closed]

So $\log^2(x) = \log(x) \log(x)$ and can be simplify with other part of function, but what $\log_3(x)$ could be develop, and $x^{\log(\log(x))}$.
1 vote
1 answer
55 views

Prove the inequality contains reciprocal fraction [closed]

I would be grateful for any tips or solution for this fractional inequality! $(\frac{a}{b})^2+(\frac{b}{c})^2+(\frac{c}{a})^2\geq\frac{b}{a}+\frac{c}{b}+\frac{a}{c}$
2 votes
0 answers
39 views

General solution for $f(n)$ satisfying $(x^n+y^n)^2=[x^2+y^2-2xyf(n)]^n$

I'm have being studying the following equation for a while; $$(x^n+y^n)^2=[x^2+y^2-2xyf(n)]^n$$ what makes this equation interesting for me is that at $n=2$ the equation can be satisfied if $f(n)=0$. ...
1 vote
1 answer
55 views

Prove that $\frac{ab+cd}{da+bc}+\frac{da+bc}{ab+cd}\geqslant\frac{4(a+c)(b+d)(ac+bd)}{2(a^2c^2+b^2d^2)+3(ab+cd)(da+bc)}$

The source is this AoPS question (unsolved). Let $a$, $b$, $c$, $d>0$, show that \[\frac{ab+cd}{da+bc}+\frac{da+bc}{ab+cd}\geqslant\frac{4(a+c)(b+d)(ac+bd)}{2(a^2c^2+b^2d^2)+3(ab+cd)(da+bc)}.\] If ...
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3 votes
0 answers
45 views

$a \le b \le c$ and $\gcd(a,b,c) = 1$ such that they are side lengths of a triangle one of whose angle is $60^\circ$. [duplicate]

Determine all triples $(a,b,c)$ of positive integers with $a \le b \le c$ and $\gcd(a,b,c) = 1$ such that they are side lengths of a triangle one of whose angle is $60^\circ$. Here if we assume that ...
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1 vote
1 answer
24 views

Isolating variable in exponents in a linear combination of exponential functions

I'm doing a physics problem that's asking me to compare the maximum speeds of a simple harmonic oscillator versus one immersed in a fluid that's leading to overdamping. I am at a point where I have ...
1 vote
1 answer
88 views

Which solution is correct: $\lim_{x \to \infty} {\frac{x^2+2x+3}{2x^2+x+5}}^\frac{3x-2}{3x+2}$?

I was solving this question $\lim_{x \to \infty} {\frac{x^2+2x+3}{2x^2+x+5}}^\frac{3x-2}{3x+2}$. Now, this is solved using $\lim_{x \to a} [1+f(x)]^\frac{1}{g(x)}=e^{\lim_{x \to a} \frac{f(x)}{g(x)}}$....
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3 votes
1 answer
82 views

Is this contraposition proof complete?

I want to prove the statement If $a>b$ then $X.$ Suppose that I've proved that If $\neg X$ then $a<b.$ This gives $a\leq b,$ which $a<b$ is a subset of. Can I consider the proof finished, by ...
2 votes
1 answer
72 views

Why can I rewrite this term as a root, but not the other?

I was practicing u-substitution. With the first problem, I was able to rewrite $u^{1/3}$ as the cube root of $u$, but when I did the same approach again with $u^{3/2}$ as the square root of $u^3$, ...
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2 votes
1 answer
73 views

Solving sextic polynomials with modular forms.

A very long time ago, I made a since removed post about solving higher order polynomials. This post demonstrated the radical solutions for degrees $1$ to $4$ and also some solutions for higher degree ...
1 vote
1 answer
37 views

What's the fourth step in this word problem about a pawn-shop?

What does "a profit of 80 percent on its buy-back price" mean? A clock store sold a certain clock to a collector for 20 percent more than the store had originally paid for the clock. When ...
0 votes
1 answer
26 views

Minimize the cost of constructing a rectangular jewelry box

Sandy is making a closed rectangular jewelry box with a square base from two different woods. The wood for the top and bottom costs $\$20/m^{2}$. The wood for the sides costs $\$30/m^{2}$. Find the ...
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-4 votes
1 answer
68 views

Computing $\lim_{x \to 0} \frac{\sin x + \sin3x + \sin5x}{\sin2x + \sin4x + \sin6x}.$ [closed]

Currently stuck on this question, would appreciate some help. I need to compute the limit of this function without using L'Hopital's rule. $ \lim_{x \to 0} \frac{\sin x + \sin3x + \sin5x}{\sin2x + \...
1 vote
1 answer
55 views

Polynomial and Sign

Suppose I have a polynomial $f(x)=0$ as $f(x)=1 + 2x - 3x^2 - 8x^3$ I can write this as $1+2x-3x^2 - 8x^3=0$. Suppose I want to write the coefficient of the highest power in positive term, so I write ...
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-2 votes
0 answers
32 views

Can have the same meaning Less than or equal and less than for natural numbers [closed]

Is this statement True for all natural numbers? $$\forall x \in \mathbb{N}, x < 11 \equiv (\forall x \in \mathbb{N}, x \leq 10) $$ Since natural numbers do not include rational numbers, can this be ...
3 votes
1 answer
72 views

Finding the height of a cone in terms of $R$ and $θ$

The problem is: A right circular cone is made from a circular piece of paper of radius $R$ by cutting out a sector of angle $\theta$ radians and gluing the cut edges of the remaining piece together. ...
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-3 votes
0 answers
27 views

Translating Trigonometric Functions [closed]

A Ferris wheel has a diameter of $17$m. You board at the bottom of the Ferris wheel from a platform $2$m off the ground. If it takes $25$ seconds to reach the top, determine an equation for the height ...
4 votes
0 answers
83 views

Find the minimum of this radical expression

For $x\in[0,6]$, find the minimum of $m=x+\dfrac{96-16x}{3\sqrt{x^2+16}}$. I used derivatives and found that the $x$ which minimizes $m$ is the positive root of \[9x^6+432x^4-2304x^2-49152x-28672=0.\]...
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2 votes
0 answers
25 views

Finding the optimal area for fitting non-overlapping squares of total area $n$ [duplicate]

Suppose we have some finite amount of squares whose total area is $n$. What is the smallest number $A$ we can find such that we can always fit the squares (no matter how many there are) with no ...
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0 votes
1 answer
55 views

Proof of $\sqrt a -\sqrt{a-1} <\sqrt{a-2} -\sqrt{a-3}$ for $a\ge3$

Given that $a\ge3,$ prove that $\sqrt a -\sqrt{a-1} <\sqrt{a-2} -\sqrt{a-3}.$ To prove this inequality, do I square both sides of this inequality? I tried to assume that $a$ is $3,$ but I don't ...
2 votes
2 answers
116 views

Do all real cubic roots have real closed forms?

This question is the same (or similar) as this one but in that question, the answer "use the trigonometric method" is not explained further in the Wikipedia page (it was 6 years ago so the ...
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-5 votes
0 answers
39 views

How do I calculate the number of hours I need to allocate for studying based on my previous grades? [closed]

I have an exam broken down by subject and percentage of total score. I am trying to see out of 500 hours, how many hours I should spend on each subject. For example: My subjects are math, reading, ...
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1 vote
1 answer
28 views

Find $3 \log(\sqrt{x}\log x) = 56, \log_{\log (x)}(x) = 54$ in base $b$

Solve the following: In a Martian civilization, all logarithms whose bases are not specified are assumed to be base $b$, for some fixed $b \geq 2$. A Martian student writes down \begin{align*}3 \log(\...
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0 votes
0 answers
40 views

What is the volume of the cuboid?

A cardboard box company uses a machine that creates cuboid (rectangular prism) boxes with outer dimensions (meaning the dimensions of the box if you are measuring from the outside of the box) of $(x+...
0 votes
2 answers
85 views

Evaluate $\int_0^{\frac{1}{2}}(1-2 x) \ln (\tan (x)) d x$

Evaluate $$\int_0^{\frac{1}{2}}(1-2 x) \ln (\tan (x)) d x$$ My try is using Feyman's Trick: $$\begin{aligned} & f(a)=\int_0^{\frac{1}{2}}(1-2 x) \ln (\tan a x) d x \\ & f^{\prime}(a)=\int_0^{...
6 votes
5 answers
155 views

Why is % difference of aggregation higher than individual % differences?

I'm calculation the % difference in completion rate (before vs after) of Product A (0.3%), Product B (16.7%) and the combination of Product A and B (17.0%). I'm unable to explain why the combination ...
2 votes
3 answers
60 views

Finding the total number of objects(stones).

I have $N$ stones. Then the stones are arranged in ascending order of weights. If I remove three stones that are heaviest, then the total weight of the stones decreases by $35$%. Now if I remove the ...
  • 131
0 votes
0 answers
45 views

If $x^3=x+1$ then $x^{3n}=a_nx+b_n+c_n/x$, where $a_{n+1}=a_n+b_n$, $b_{n+1}=a_n+b_n+c_n$,$c_{n+1}=a_n+c_n$.

If $x^3=x+1$ then $x^{3n}=a_nx+b_n+c_n/x$, where $a_{n+1}=a_n+b_n$, $b_{n+1}=a_n+b_n+c_n$,$c_{n+1}=a_n+c_n$. I am not sure how to solve this, it seems like this statement is true for $x^3$ and $x^6$ ...
0 votes
1 answer
48 views

question about the algebra in finding the range of $\tan$

This is part of a question about the range of tan. The algebra is tripping me up. Let $P = (x, y)$ be the point on the unit circle that corresponds to an angle $t$. Consider the equation $\tan(t)= \...
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0 votes
1 answer
48 views

Showing that $\sup_{x\in[0,1]}|f(x)|\leq \sqrt{\int_0^1(f'(x))^2dx}$ when $f\in C^1([0,1],\mathbb{R})$ and $f(0) = 0$

Let $f\in C^1([0,1],\mathbb{R})$ such that $f(0) = 0$. I am trying to show that $\sup_{x\in[0,1]}|f(x)|\leq \sqrt{\int_0^1(f'(x))^2dx}$. I have a suspicion that we ought to use the mean value theorem ...
4 votes
2 answers
205 views

Boxes and dice game

You have $9$ boxes numbered $1$ through $9$ and then you have two $6$-sided dice. Each turn you roll the two dice and deduct the sum of the dice from the boxes by removing the number itself or any ...
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3 votes
1 answer
84 views

Generate a set of numbers that have uniqueness on addition, subtraction, multiplication and division

The problem is somewhat tricky. I would like to generate a set of number pairs $(x_1, y_1), (x_2, y_2)...$ so that each pair of numbers has uniqueness on all four computations (addition, subtraction, ...
0 votes
1 answer
40 views

Factoring out an expression to reach the answer

Can you tell me what is wrong of my solution that is not found in possible answers which are $1, {{5}{/}{2}}, \sqrt{2}, 2, 3 $ $$ \text { if } \quad x+\frac{2}{\sqrt{x}}=5 \quad \text {, what is} \...

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