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.

Filter by
Sorted by
Tagged with
0
votes
1answer
47 views

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
0
votes
1answer
17 views

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 ...
2
votes
2answers
64 views

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 ...
1
vote
1answer
23 views

How to isolate y in a potential function

The following is a potential function: $Φ(x,y)=x^4y^4+x^2e^xy^2-3lnx-\frac{3}{4}e^2=0$ I would like to isolate $y$ to give $y(x)=…$. An online calculator gave $y=\pm \frac{\sqrt{-\sqrt{x^4(e^2x+12ln(...
1
vote
2answers
59 views

Does this rule I found really work?

I was playing a bit whit exponents. I maybe found a working formula for calculating $n^y$ if you know $n^x$. The formula may already be discovered, but the formula I found is: $$ (n^x)^\frac{y}{x} = n^...
0
votes
5answers
63 views

How to Solve for $x$ in a Particular Exponential Equation

I am trying to solve for $x$ in $x^2=(16)^{2x}.$ So I started this way: I took square root of both sides and got $x=16^x$ Then I took the logarithm of both sides and got $\log x=x \log 16.$ This ...
0
votes
2answers
38 views

Division algorithm proof equality

On the following website there is a statement: $$a - ba = -a(b-1)$$ $a$ is negative and $b$ is positive. How does this rearrangement happen? I see that it is equal, but I can not see how to ...
0
votes
1answer
38 views

Show that $x<x^n<1$ when $x,n\in(0,1)$

Show that $x<x^n<1$ when $x,n\in(0,1)$ My attempt: If I could show the function $x^n-x$ is positive in the given interval... $x^n-x = x(x^{n-1}-1)$ Since $x$ is positive we can ignore it and ...
2
votes
1answer
58 views

If $x,y,z\in\mathbb{R}^+$, prove that $\sqrt{x^2-xz+z^2}+\sqrt{y^2-yz+z^2}\ge\sqrt{x^2+xy+y^2}.$ [duplicate]

When I was doing Math Training, the coach gave a inequality problem. If $x,y,z\in\mathbb{R}^+$, prove that $$\sqrt{x^2-xz+z^2}+\sqrt{y^2-yz+z^2}\ge\sqrt{x^2+xy+y^2}.$$ I tried to use the brute ...
-1
votes
3answers
47 views

Binomial expansion of negative exponent

Stuck on this binomial result doing gravitation chapter in physics the expression is $$ (1+x)^{-2}=1-2x $$ provided $x$ is so smaller than $1$ . My questions is Why and second If I want to ...
-2
votes
0answers
46 views

Question regarding logarithm [on hold]

Solve for $n$ : $$n^2 2^n = 2000$$ Thanks!
0
votes
2answers
54 views

Algebra and Combinatorics books for Mathematical Olympiads

Could you kindly point out to me, some good contest-preparation book's to develop theory and problem solving skills in Algebra? It would be good if the book is less of theory and more of problems. I ...
0
votes
3answers
39 views

Factor negative from quadratic equation

$-4x^2+20x-25=0$ Then you factor out -1 $-1(4x^2-20x+25)=0$ Then factor again into $(a-b)^2$ $-1(2x-5)^2=0$ In the second equation, can I multiply both sides by -1, why did the author leave "-1" ...
2
votes
5answers
67 views

How to solve $\log_{3}(x) = \log_{\frac{1}{3}}(x) + 8$

I'm trying to solve $\log_{3}(x) = \log_{\frac{1}{3}}(x) + 8$. I am getting x = 4 but the book gets x = 81. What am I doing wrong? \begin{align*} \log_{3}(x) & = \log_{\frac{1}{3}}(x) + 8\\ \...
1
vote
2answers
33 views

Confused with a system of equations with three variables that has infinitely many solutions

I'm studying High School Algebra and it had this question: Solve the system by equations: \begin{align*} x + y - z &= \,0 \\ 2x + 4y - 2z &= 6 \\ 3x + 6y - 3z &= \,9 \end{...
1
vote
3answers
48 views

How to solve equation involving ceiling function

I am trying to show the following holds when $n$ is not divisible by $1+\Delta$. $⌈ \frac{n}{1+\Delta} ⌉ = n- \Delta \implies \Delta=n-2$ I have tried using the Quotient Reminder Theorem but I am ...
3
votes
2answers
31 views

Does order matter when subtracting systems of linear equations?

Let's say I have a simple system like so: \begin{cases} 2x - y = 1\\ x - y = 2\\ \end{cases} Why does it not matter what equation is being subtracted from the other when it comes to getting ...
0
votes
2answers
43 views

Derivative of an Inverse, Can't find Inverse

Taken from a single variable calc book. Find $g'(a)$, where g is the inverse function of the given function $f(x)=x^5-x^3+2x, a=2$ I intend to use the formula $g'(a)=\frac{1}{f'(g(a))}$, and I know ...
0
votes
1answer
16 views

Combined variation with four variables: Is this even possible to solve?

I have this problem, and I'm quite confused on how to solve it. Mr. Plaridel owned a newspaper publication called Diaryong Tagalog Inc. He observed that when he used 3 printing presses, he can ...
2
votes
0answers
39 views

Complicated inequality proof

I've run into a stumbling block as part of a larger problem where I need to show if $$\sum_i w_i = 1$$ This implies $$ \frac{\sum_j w_j^2( (1+w_j^2) +\sum_iw_i^2) }{\sum_i w_i^2 +1} \leq 1$$ I'm ...
0
votes
1answer
29 views

Algebra, Relationship between multivariable functions with knowns and unknowns

Suppose $f$ is a function of two variables, $x$ and $y$. where $x, y$, and $f(x,y)$ are all real numbers. The question is if there is $x_1$ and $x_2$, known numbers, what do we know about the ...
-1
votes
1answer
48 views

What is the sum of all values $p$ satisfying this equation. [on hold]

Consider the following equation: $$\frac{n^3-12n^2+8n+93}{11+2n-n^2} = p$$ Let $n$ be some integer satisfying above equation yielding a prime number $p$. What is the sum of all possible values of $...
0
votes
0answers
37 views

Expression evaluation/Book

I have a problem with deriving the second displayed formula in the snippet (which comes from the book Game Theory by Tirole and Fudenberg): in the square brackets should be $$\Bigg [\frac{3(1-\delta)+...
-2
votes
4answers
49 views

What's wrong in my argument? [on hold]

The limit $\lim_{n \to \infty}\left( \frac{3^n}{2^n}\right)=\infty $ might be rewritten as $\lim_{n \to \infty}\left( \frac{3}{2}\right)^n=\infty.$ But what if $\frac{\ln(3)}{\ln(2)}$ instead? Let’...
1
vote
4answers
50 views

Let $f (x$) = $\sqrt{−x^2 + 20x + 400} + \sqrt {x^2 − 20x}$. How many elements in the range of $f$ are integers?

Let $f (x$) = $\sqrt{−x^2 + 20x + 400} + \sqrt {x^2 − 20x}$. How many elements in the range of $f$ are integers? I first let $y= x^2 -20x +100$. Then substitute it in the function -------> $f(x) = ...
0
votes
2answers
28 views

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 ...
0
votes
1answer
38 views

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 ...
0
votes
4answers
98 views

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?
0
votes
2answers
23 views

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 ...
6
votes
3answers
179 views

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....
-1
votes
1answer
49 views

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 ...
1
vote
0answers
50 views

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, ...
3
votes
4answers
59 views

Solving $\tan\theta=\cos\theta$, for $-\pi <\theta < \pi$

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\...
1
vote
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 ...
0
votes
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 ...
4
votes
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^{-...
3
votes
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 ...
1
vote
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. ...
0
votes
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 ${\gamma}^2$, ...
1
vote
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, ...
0
votes
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 ...
1
vote
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): $\...
2
votes
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 ...
3
votes
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 $$ \...
1
vote
1answer
90 views

Solving simple equation

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 ...
1
vote
1answer
74 views

On the greatest integer that cannot be expressed as $c_{1}a_{1}+c_{2}a_{2}$

Let $G(a_{1},a_{2})$ be the greatest integer that can not be expressed as $c_{1}a_{1}+c_{2}a_{2}$, where $a_{i}$'s are relatively prime natural numbers, and $c_{i}$ is a whole number. Formula: $G(...
2
votes
3answers
71 views

Logarithmic inequality (looking for a better solution)

$1+\sqrt{17-\log_{x}{2}} \cdot \log_{2}{x^7} \geq \log_{2}{x^{27}}$ Let $t = \log_{2}{x}$. Then we get (taking account of the fact that $x>0$ and $x \ne 1$ $$1+\sqrt{17-\frac{1}{t}}\cdot 7t \geq ...
0
votes
1answer
49 views

Is this true that this equation has three possible roots? [on hold]

Is this true that this equation has three solutions (provided $f(x)$, $g(x)$ are defined): $f(x) = 0 $, provided $g(x)$ is defined, or $g(x) =0 $, provided $f(x)$ is defined, or $f(x)+g(x)=0 $, ...
0
votes
1answer
37 views

Mathematics of Growth

I wondered if anyone could help with the relevant steps for particular problem below. Suppose for example, world population was about $679$ million in the year $1700$ and $954$ million in $1800$. I ...
0
votes
1answer
35 views

Resolve into partial fractions $(x^2 + 3x - 5)/[(2x - 7)(x^2 + 3)^2]$

Resolve into partial fractions $\frac{x^2 + 3x - 5}{(2x - 7)(x^2 + 3)^3}$ The question has to do with the denominator being one linear and a repeated quadratic factor. Although, I am familiar with ...