# 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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### How do i find the inverse of this function?

$$y=\frac{1}{(256-x) \cdot 0.5^x}$$ I can't seem to find a solution anywhere and don't know how to go about it
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### Simple Algebraic Problem: Help with Soccer Team Trip Problem (arithmetic)

An easy one but I don't know whether the result is correct (decimals). A soccer team wants to celebrate their victory and they decide to go on a trip. The team is made up of 20 students The trip is ...
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### Solve $\frac{8^x+27^x}{12^x+18^x}=\frac{7}{6}$ for $x\in\mathbb{R}.$

Solve $$\frac{8^x+27^x}{12^x+18^x}=\frac{7}{6}$$ for $x\in\mathbb{R}.$ There is a brute force method that relies on breaking each exponential up into $2^x$’s and $3^x$’s and substituting, but I am ...
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### Find the length of the race track

A cycle track is a right triangle with a difference of 2 km between the legs.If the hypotenuse passes along a side road and the two legs pass along a highway.One of the participants of the cycle race ...
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### Ceiling and Floor Function Properties

When solving a problem I came across the expression $⌈n⌉+⌊m⌋$, where n and m are real numbers. Given that we know: $n≤⌈n⌉<n+1$ and $m-1<⌊m⌋≤m$, I was wondering if any conclusions can be drawn ...
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### Find $z^5 +\frac{1}{z^5}$ given that $z > 0$ and $z^2 + \frac{1}{z^2}=14$

I have been struggling with this problem and I think I am nearly there. Here is my working so far: Any hints?
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### How to factor this expression in the fireman's problem?

I'm trying to solve the fireman's optimization problem. It boils down to factoring the following expression: $2\left ( q + \frac{pq}{a} \right )\cdot \left ( -\frac{pq}{a^2} \right ) + 2(p+a)$ I ...
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### Rearranging the formula

Transpose this formula to make $y$ the subject. $$x=\sqrt{x^2y^2+1-y}$$ My try: $$x^2=x^2y^2+1-y$$ $$x^2-x^2y^2=1-y$$ $$x^2(1-y^2)=1-y$$ Here I got 2 $y$ terms, but I am not sure what to do next....
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### finding roots of an equation [on hold]

I want to find the smallest positive root of equation: $$x^3-0.75x+b=0$$ when $$b = \frac{1}{64}\left[(\sqrt{5}-1)(\sqrt{6}+\sqrt{2})-(\sqrt{6}-\sqrt{2})\sqrt{10+2\sqrt{5}}\right]$$ I know value of ...
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### Paper about “where is the knee of an exponentional function?”

It is a common mistake to name some point of an exponential function "knee". Some years ago I found a nice paper which collected some examples and some extra anecdotes about this. I do not remember, ...
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Solve for $-\pi <\theta < \pi$: $$\tan\theta=\cos\theta$$ I can't get to the correct solution using the identities: $$\tan\theta=\frac{\sin\theta}{\cos\theta} \quad\text{and}\quad \sin^2\... 2answers 50 views ### Finding all the solutions to \sin(5x) - \sin (3x) = \sqrt 2 \; \cos(4x) I had to solve$$\sin(5x) - \sin (3x) = \sqrt 2 \; \cos(4x)$$After working with the equation, I got$$2\sin(x)\cos(4x) = \sqrt 2 \; \cos(4x)$$with difference of sines formula. I saw that, I ... 0answers 41 views ### An interesting question on quadratic equations in which all coefficients are unknowns and only the range in which the roots lie is been given [duplicate] Consider the quadratic equation ax^2-bx+c=0, where a,b,c are natural numbers. This equation has 2 distinct, real roots lying in the interval (1,2). The question is to find the minimum value of ... 2answers 68 views ### What did I do wrong with this logarithmic equation?$$e^{3x-2}e^{-x}=4e, \ \text{round to the nearest thousandths}I keep getting x\approx2.884 but the answer is x\approx2.193. What am I doing wrong? Here is my work: \begin{align*} e^{3x-2}e^{-... 2answers 49 views ### Problem: Associative law and the use of parentheses (Algebra) this time I have a problem with parentheses, The problem in question I transcribe it in case you can't see it properly: How many "(" and ")" symbols do you need to specify completely the order of ... 1answer 20 views ### how to find max profit using demand and cost so i have been stuck on this question for a while. It asks you to find out how much of each product you need to produce in order to be able to maximise profits. Now i know that Profits=Revenue-Cost. ... 2answers 30 views ### Find a new cubic equation with new roots \alpha\beta, \beta\gamma and \gamma\alpha. Can I use substitution for this? Given a cubic equation x^3+2x^2-5x+1=0, find the equation with roots \alpha\beta, \beta\gamma and \gamma\alpha. For the case where the new roots are {\alpha}^2, {\beta}^2 and {\gamma}^2, ... 3answers 46 views ### Intersection between y=x and y = \ln (x-2) + b There is no context relating to this question or any previous information given. The question is just as follows: The line y=x is a tangent to the curve \ln(x-2)+b. Using calculus or otherwise, ... 1answer 22 views ### two circles on different cycles I am not a math person but I am going to try and phrase this according to the rules. You have an object that completes a path every 29,000 years (earth orbit), that object also has a tilt that is on a ... 2answers 46 views ### Finding the equation of a rational function from a set of points I want to fit a curve onto a set of points. The curve should look like a log or square root function.I want the function to be specifically in this format (per the requirements of my supervisor): \... 2answers 51 views ### Two taps are filling a tank [on hold] Two taps are filling a tank together in 60 minutes (each tap has a different filling speed). If we open only one tap until we fill 1/3 of the tank, then close it, and then let the other tap to fill ... 1answer 38 views ### Iterated Integral With Odd Upper-Bound Let f:[0,1]\rightarrow \mathbb{R} be an Lebesgue integrable function on [0,1]. I would like to compute the following integral but I believe that I may be wrong when applying Fubini's theorem \...
I can't seem to solve/simplify step by step to get from equation 3) to 4) as they do in this paper. As per the paper: 3) $p = \frac{p' + (r-R)p}{1+r}$ Because both sides of equation 3) involve ...