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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.

0
votes
1answer
32 views

Problem with finding solutions to polynomial equation

What is the best method to find the solutions to this equation? $x^3(x^2+2)=-297$ I have obtained the solution from the back of the book but have no idea as to the method to get the solution. Is it ...
1
vote
0answers
31 views

When is $a(n)$ prime?

Question: When is $a(n)\in P$ compared to all possible values of $n$? where $P$ denotes the set of primes. What is the density of the primes in the sequence? Consider the sum of the prime counting ...
0
votes
1answer
44 views

Finding $n$ such that $n^2 + 2n + 2019$ is a perfect square

What $n$ solves $n^2 + 2n + 2019$ for the expression to be a perfect square?
1
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4answers
73 views

If $i^2=-1$ why isn't $i$ also equal to $-\sqrt{-1}$?

I've read on this site several times that it is better to say $i^2=-1$ as opposed to $i=\sqrt{-1}$. If we let $i^2=-1$, why doesn't $i=-\sqrt{-1}$? This makes sense since $(-\sqrt{-1})(-\sqrt{-1})=-...
0
votes
1answer
35 views

Formal definition of “ inverse operations” on a set?

I'd like to express formally the fact that multiplication and division are inverse operations. In order to do that, I'd like to find a general definition of " inverse operations" on a set. Suppose ...
1
vote
5answers
53 views

Find value of n if ${(n-1)! \over (n-3)!} = 30$

So here's my attempt : $${(n-1)(n-2)(n-3)! \over (n-3)!} = 30$$ $${(n-1)(n-2)} = 30$$ then $$n-1 = 30$$ or $$n-2 = 30$$ then $$n = 31$$ or $$n = 32$$ but my textbook says $$(n-1)(n-2) = 6 * 5$$ ...
0
votes
0answers
15 views

Proving an inequality involving absolute value; how do I justify using a conjunction (and) instead of a disjunction (or)?

I'm putting together the following the proof, and I have a question about one of the final steps. Definition of absolute value: $\forall x \in \mathbb{R}, (x \geq 0 \Rightarrow |x| = x) \...
2
votes
2answers
225 views

Solving this system of quadratic equations [on hold]

I'm having troubles solving the following system. $$\begin{aligned} (2x-y)^2 -3xz &= 0\\-3(2x-y)^2-8yz &= 0\\-9x^2-4y^2+36 &= 0 \end{aligned}$$ I tried by elimination, but I have ...
0
votes
1answer
20 views

Given an inequality of the form (x-a), (x-b), how do the factors (x-a), (x-b) split the number line into the following 3 parts?

I encountered this statement on the U of T math department website (under the first example): In general, if you want to solve an inequality of the form $(x-a)(x-b) > 0$ [...], notice that the ...
3
votes
1answer
66 views

Find a remainder when $(x^5+1)^{100} + (x^5-1)^{100}$ is divided by $x^4+x^2+1$

The question is: Find a remainder when $f(x)=(x^5+1)^{100} + (x^5-1)^{100}$ is divided by $x^4+x^2+1$ I first began by decomposing $$x^4+x^2+1=(x^2+x+1)(x^2-x+1)$$ and using $$x^3-1=(x-1)(x^2+x+1)\...
-3
votes
2answers
33 views

All results of the equation x∈R [on hold]

This is the equation; $\ln(x+1)+\ln(x)=\frac{1}{2}\ln(49)$ I can't solve it could somebody help me and show me the way thx a lot.
2
votes
0answers
54 views

Induction Proof: Define the sequence $ \ \ T_1, T_2, T_3,…\ \ $ by $\ \ T_1=T_2=T_3=1 \ \ $ and $ \ \ T_{k+1}= T_{k}+T_{k-1}+T_{k-2}$.

I would appreciate if someone could comment my solution of the task below: Define the sequence $ \ \ T_1, T_2, T_3,...\ \ $ by $\ \ T_1=T_2=T_3=1 \ \ $ and $ \ \ T_{k+1}= T_{k}+T_{k-1}+T_{k-2}$. ...
1
vote
1answer
68 views

6th degree equation [duplicate]

The number of real roots of $$ \frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19} = x^2 -11x -4 $$ How to solve without actually finding the roots or is it the only way? I know that if the ...
0
votes
2answers
40 views

Сonvert trigonometric equation to quadratic equation

I am struggling to know how to transform this equation as it involves more than one type of trigonometric function, I know how to do it with one repeated function. Question: $\sin^2 \theta/2$ + $\...
0
votes
1answer
28 views

Calculate initial velocity based on displacement, time and constant acceleration.

"A car has a constant speed along a road. It goes down a hill at a constant acceleration. 50s after it goes down the hill the speed is doubled and 50s later it reaches the end of the 200m hill and is ...
0
votes
2answers
9 views

What proportion of 27k should both individuals pay?

I have 2 individuals one making 177 the other making 415, what percentage of 27 should each pay to make it proportional to their incomes? I believe the setup is the following: ...
1
vote
4answers
24 views

Complex number cannot arive at $\frac{9}{2}-\frac{9}{2}i$ with problem $\frac{4+i}{i}+\frac{3-4i}{1-i}$

I am asked to evaluate: $\frac{4+i}{i}+\frac{3-4i}{1-i}$ The provided solution is: $\frac{9}{2}-\frac{9}{2}i$ I arrived at a divide by zero error which must be incorrect. My working: $\frac{4+i}{i}$...
2
votes
2answers
46 views

How can I solve this problem using a formula?

Four friends ate $7$ different dishes in $5$ minutes. If they were joined by $X$ people the next day and ate at the same rate, how many dishes would they eat in $Y$ minutes? For example if $X = 4$ ...
0
votes
0answers
22 views

Need help with the integral: $\int_{0}^{\infty}\prod_{i}^{N/2}(e^{-\lambda_{2i-1}t}+e^{-\lambda_{2i}t}-e^{-(\lambda_{2i-1}+\lambda_{2i})t})dt$

I am trying to understand the behavior of the following integral $$\int_{0}^{\infty}\prod_{i=1}^{N/2}\left( e^{-\lambda_{2i-1}t}+e^{-\lambda_{2i}t}-e^{-(\lambda_{2i-1}+\lambda_{2i})t} \right)dt$$ ...
3
votes
3answers
38 views

Complex Number: $\frac{(3+i)^2}{(1+2i)^2}$ - cannot arrive at textbook solution

I have a complex quotient $\frac{(3+i)^2}{(1+2i)^2}$ The solution provided in my textbook is $-2i$. I arrived at different solutions and I'd like to know where I went wrong. Till now in my textbook ...
1
vote
1answer
39 views

10th Grade Algebraic Rate of Bacteria Growth Problem

"A certain type of bacteria doubles every 6.5 hours. If there were 60 bacteria to start with, what is the hourly growth rate of the bacteria? How many bacteria will there be after a day and a half? ...
-1
votes
1answer
26 views

Separable Differential Equation without x

5y'= y(y+5) Hi, I'm blocked on this differential equation. The book suggests the separation method, but I don't know how to get the general equation in this specific case. Thanks to all. It's my first ...
1
vote
0answers
45 views

Numbers of way to solve quadratic equation

How many ways can a quadratic equation of $ax^2+bx+c=0$ be solved? The most common one I know is the quadratic formula and factorization method. $x=\frac{-b+\sqrt{b^2+4ac}}{2a}$ and $(ax+b)(cx+d)=0$...
0
votes
2answers
30 views

Question relating to logarithms

Hello I have the following question I have to find the answer of this logarithmic formula $$(\log(2^5) + \log(4^{0.2}))\times(\log(5^2) + \log(25^{0.5})) .$$ I am currently having problems with the ...
-2
votes
0answers
31 views

Symmetry in the roots of a quadratic [on hold]

Given $m$ is a root of $x^2+ax+b=0$ find all the possible values of $(a,b)$ such that $m^2-2$ is also a root.
0
votes
3answers
119 views

Factorise $1+x^2$

How do I factorise this expression? $$1+x^2$$ An attempt: complete the square $(1-x)(1+x).$ teacher said no. $x(1/x+x)$ again teacher said no. She said is related to solving this $x^2+1=0$. I ...
0
votes
0answers
31 views

System of algebraic equations concerning rates

I'm not even sure where to start on the problem. Is this possible to solve without logarithms? "A certain type of bacteria doubles every 6.5 hours. If there were 60 bacteria to start with, what is ...
1
vote
0answers
22 views

How many nodes are there of each degree?

Q:A graph has 12 edges and 6 nodes, each of which has degree of 2 or 5. How many nodes are there of each degree? let m = the amount of edges, then $$ 2m = \sum_{v\in V} deg(V) $$ from here we can ...
1
vote
1answer
47 views

Finding the Minimum Value of Sum of Positive Real Numbers

I'm not sure how to solve the following problem: $$d_1^2 + \ldots + d_n^2 = \sigma^2$$ $$d_i \geq 0$$ Find the minimum possible value of $$d_1 + \ldots + d_n$$ I have a hunch that its when every ...
0
votes
1answer
36 views

Need help with this word problem using hyperbolas and need the final answer in (x,y)

John was in the lead, Ed was 1.5 miles behind and Jeff was 2 miles behind John. Then they heard an explosion. John heard it first and Ed heard it a second later and Jeff heard it 1.5 seconds after ...
0
votes
3answers
50 views

given $x^2 + y^2 = 2x$. I want $(x-1)^2 + y^2 = 1$

given $x^2 + y^2 = 2x$. I want $(x-1)^2 + y^2 = 1$ Is this completing the square? my attempt: $$x^2 + y^2 - 2x = 0$$ need $1$ $$x^2 + y^2 - 2x + 1 = 1$$ I forgot how to could someone explain ...
4
votes
1answer
44 views

$\cos(n\vartheta)=\frac{a_n}{3^n}$

I want to show, if i know that $\cos(\vartheta)=\frac{1}{3}$ than $\cos(n\vartheta)=\frac{a_n}{3^n}$ for $n\in \mathbb{N}$, where $a_n \in \mathbb{Z}$,$3 \nmid a_n $ My approach was to do it by ...
0
votes
3answers
71 views

$m-m\sqrt{m}=80$ $\Rightarrow m- 5\sqrt{m}=?$

$m-m\sqrt{m}=80$ $\Rightarrow m- 5\sqrt{m}=?$ I tried different approaches such as, substituting $\sqrt{m}=t$, squaring the given equations, substituting $\frac{m-m\sqrt{m}}{16}$ as $5$. Nothing ...
0
votes
3answers
26 views

Finding the ratio of 5th of two different arithmetic sequences

It is given that the ratio of the sum to the nth term of two different arithmetic sequences is $7n+2:n+3$. Find the ratio of the 5th term of the sequences. I have no idea where to start this pls help!
-3
votes
1answer
38 views

Floor function linear problem [on hold]

Solve $\lfloor\frac12x\rfloor +\lfloor\frac23x\rfloor=x$, where $x$ is a real number. Is there any method besides trial and error???
3
votes
3answers
60 views

find the maximum value of $xy + yz +zx$ [duplicate]

find the maximum value of $$xy + yz +zx$$given that $x+2y+z=4$ my attempt : $(x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx) $ or $2S=2(xy+zx+zy)=(x+y+z)^2 -x^2-y^2-z^2=(4-y)^2-x^2-y^2-z^2$ $2S=-x^2-z^2-8y+16=-...
0
votes
1answer
40 views

What constant $c$ will make $ \sum_{k=2}^{N}c^{\frac{1}{k\log k}}=N ?$

What constant $c$ will make this equality valid for any $N$ chosen? $$ \sum_{k=2}^{N}c^{\frac{1}{k\log k}}=N. $$ I tried getting a rough idea of what $c$ should be and got about $1.46$ when $N=1000$ ...
0
votes
3answers
39 views

Check if true : Atleast one of the integers, a, b, c must be even

Suppose a, b, c are integers such that the equation $ ax^2 + bx + c =. 0 $ has a rational root. Check if true : Atleast one of the integers, a, b, c must be even. I know for rational roots $ b^2 - ...
0
votes
1answer
42 views

For how many real numbers 'b', does $x^2 + bx + 6b = 0$ have one integral root?

The question states. For how many real numbers 'b', does $ f(x) =x^2 + bx + 6b = 0$ have one integral root ? My line of thinking : Let $\alpha , \beta$ be the roots of of $f(x)$. $\alpha + \beta = -b....
1
vote
0answers
69 views

How to find this area of this figure without using calculus?

Is it the same as the sector of Angle $\theta$ ? Can you find the shaded area for $\theta=90$ degree ?
0
votes
0answers
22 views

Let $f(x)=(x−x_1)(x−x_2)(x−x_3)(x−x_4)$ such that $x_2>x_1$ and $x_4>x_3$. If range of $f(x)$ is $(−\infty,14]\cup(1,\infty)$ then [on hold]

A)$x_1<x_3<x_4<x_2$ B)$x_3<x_1<x_2<x_4$ C)$x_1+x_2=x_3+x_4$ D)$f'(x)=0$ has two real roots I tried this question whuch appeared in my paper but couldn't find the approach to ...
-3
votes
2answers
58 views

Number of integral solutions for the equation $x + 2y = 2xy$ is? [on hold]

How many integral solutions does the following equation have? $$x + 2y = 2xy$$ I have tried hit and trail method and I got only one solution, namley, $x=y=0$.. But Is there any other way to solve ...
0
votes
1answer
46 views

Is $\tan^{-1}(-1) = 3\pi /4$ or $=7\pi /4$? I understand they're both valid solutions, but what about places where the value is added/subtracted?

For example: calculate $\int^4_2 \tan^{-1} x \, dx$ If $\tan^{-1}(-1) = 3\pi /4$, then the final answer is $\frac{-\pi}{2}$. But if $\tan^{-1}(-1)= 7\pi /4$, then the final answer would be $\frac{3}{-...
0
votes
1answer
35 views

Distance of the point $(a,b,c)$ to the plane $z=0$

I'm trying to solve a calculus problem, I need to find the mass of a cylinder, I'm close to the answer, I got $8\pi$ but it should be $16\pi$. I think my mistake lies in the density function since it ...
0
votes
2answers
19 views

Find unknown matrix element using elementary row operation?

$A=\left(\begin{matrix}x & 5 & x \\ 1 & 3 & -2\\ -2 &-2 &2 \end{matrix}\right)$ $B=\left(\begin{matrix}0 & 0 & 21 \\ 1 & -1 & -14\\ 0 &\frac{4}{3} &...
2
votes
3answers
53 views

Find all the $z$ such that $(1+\frac{1}{z})^{4}=1$

Question if $S$ be the set of solution of $$\bigg(1+\frac{1}{z}\bigg)^{4}=1$$ then prove that the points are co-linear. Attempt $\bigg(1+\frac{1}{z}\bigg)^{4}=1$ $\implies z^4+4z^3+6z^2+4z+1=z^4$ $...
0
votes
1answer
57 views

Asymptotic growth of $\frac{k}{1}+\frac{k^2}{1\cdot 2}+\frac{k^3}{1\cdot 2\cdot 4}+\dots$

Let $k$ be a positive integer, and let $$n=\frac{k}{1}+\frac{k^2}{1\cdot 2}+\frac{k^3}{1\cdot 2\cdot 4}+\frac{k^4}{1\cdot 2\cdot 4\cdot 8}+\dots,$$ where the sum goes on until the next term in the sum ...
0
votes
1answer
19 views

Fractional Exponents Microeconomics

I need to find $X_a$ using this equation. I am having trouble working out how they got this answer. The question is $$ Y=X_a^{1/3} \left(\frac{w_a}{w_2} X_a \right)^{1/3} $$ The answer works out ...
0
votes
3answers
38 views

Factorization of $(m-n)p^3-(m-p)n^3+(n-p)m^3$ [on hold]

Factorize $(m-n)p^3-(m-p)n^3+(n-p)m^3$
1
vote
2answers
42 views

An inequality with positive integers

Let $s$ be a positive integer. Let $a_1, a_2,\ldots, a_s$ be distinct positive integers such that $a_i\geq2, \ \forall \ i\in\lbrace 1,2,\ldots, s\rbrace$. Show that $(a_1a_2\ldots a_s)^2-a_1a_2\ldots ...