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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0
votes
2answers
34 views

How can I make sense of $A+(-B)=A-B$?

I know the reason underlying it that why negative of a negative number is positive and why positive of a negative number is negative. But my query is not what it seems to be. Actually what I want to ...
-1
votes
2answers
18 views

The time (T) required to do a job varies inversely as the number of people (P) working. [on hold]

So, It takes 10 hrs for 4 workers to paint the warehouse. How long will it take 12 workers to paint the warehouse?
0
votes
1answer
26 views

Determining Exact Values of Trignometric Equations

Use the special triangles to give exact solutions where possible. Find all values of $x$ such that $0\le x \le 2\pi$ . (a) $\tan^2 x=1$ $\,$ (b)$\, \, 2\cos x + \sqrt{3}=0 \, \,$ (c) $\, \, ...
0
votes
1answer
8 views

How do I find the value in a range based on the value in another range?

For example Range1 = 0.6 to 0.7 Range2 = 0.0 to 0.5 When Range1 = 0.6(min) Range2 = 0.5(max) Inversely when Range1 = 0.7(max) Range2 = 0.0(min) So lets say the value of Range1 = 0.65, how do I ...
0
votes
1answer
22 views

Finding the area of the triangle with vertices at $(ct, c/t)$, $(-ct, -c/t)$, $(ct^{2}, 2ct)$

Question 8, part (iv) asks me to: 8. Find the area of each of the triangles whose vertices are as given below: (iv) $(ct, c/t)$, $(-ct, -c/t)$, $(ct^{2}, 2ct)$. So far (i.e. in parts (i), ...
-3
votes
0answers
25 views

Suppose you borrow $1,200 at an add-on interest rate of 5% for 4 years. What will be the amount of your monthly payment? [on hold]

Suppose you borrow $1,200 at an add-on interest rate of 5% for 4 years. What will be the amount of your monthly payment?
2
votes
2answers
30 views

Factoring a quadratic equation with complex numbers

I'm very new to complex numbers and am having some difficulty factoring a quadratic polynomial: $$x^2-2x+10.$$ Using the quadratic formula gives $$x=\frac{4 \pm\sqrt{4-2(1)10}}{2(1)}=\frac{2 \pm ...
0
votes
1answer
68 views

What is this shape ? $|x|+|y|+|z|=1$

What is this shape ? $|x|+|y|+|z|=1$ I thought it might be a sphere in R^3 but dont really know how to determine this...
0
votes
2answers
25 views

Solution to safe with six keys, largest number of people with 3 different keys each.

A safe has six locks. A group of people each receive a different set of three keys to the safe. Any two people should not be able to open the safe, due to missing at least one key. What is the largest ...
-1
votes
1answer
39 views

Proving a complex numbers result

$z$ is defined as $z=a+bi$. Show that $|z|^2=zz^*$ and $(z-ki)^*=z^* +ki$. In an argand diagram a set of points representing the complex number $z$ is defined by the equation ...
-6
votes
1answer
47 views

Prove that $3^n$ divides $(3n)!$ for all $n≥ 1$. [on hold]

Prove that $3^n$ divides $(3n)!$ for all $n≥ 1$. For example, $3$ divides $3!=6$ and $9$ divides $6! = 720$
2
votes
3answers
40 views

$x_1 + x_2 + x_3 \le 50$ solutions

The book shows the answer as attached. Their equation, $$x_1 + x_2 + x_3 + y = 50 \implies x_1 + x_2 + x_3 = 50 - y$$ How is that the same as solving, $$x_1 + x_2 + x_3 \le 50$$ ???
3
votes
1answer
54 views

How does this line came?

:) Today I started to learn algebra with my own. And the first chapter which I'm learning is Ratio. I'm kind of confuse in an example, in all the lines starting with (*). Until first three lines of ...
0
votes
1answer
16 views

comprehending the nature of multivariable equations

How to determine what equations such as $(x+1)^2+(y-2)^2=0\ $ and $x^2 +(y-1)^2+(z+1)^2=0$ determine in the $\Bbb R^3$ space? I can't tell when it's a plane and when it's a line...?
0
votes
1answer
8 views

group of points which this eqution determines

I suspect that $xy+x=0$ determines the equation of an infinite line in $R^3$ for which a general coordinate is $(0,-1,z)$ since in the end, it is the intersection of x=0 and y=-1 (the planes). is this ...
1
vote
1answer
22 views

Number of algebraic solutions to a formula related to a square tiling problem

How can many different sets of prime-factors fit together so well in this formula? I am curious about the number of solutions to the following equation: $$ r_3 = \sqrt{2}\; \frac{ 1 + r_1 (r_2 ...
2
votes
2answers
58 views

Find all $n\in\mathbb{Z}^+$ such that the sum of the digits of $5^n$ equals $2^n$ [duplicate]

Find all $n\in\mathbb{Z}^+$ such that the sum of the digits of $5^n$ equals $2^n$ Starting with a table of values, I found that $n=3$ works. Beyond this, it's hard to imagine any other number ...
0
votes
0answers
24 views

Problem about fibonacci sequence via quadratic roots in gelfand's algebra text.Need hints.

I have solved a preceding question proving that the common ratio of such a sequence is $ \frac {1+\sqrt{5}}{2} $ or $ \frac {1-\sqrt{5}}{2} $ (resolving a quadratic equation) . The present problem is ...
2
votes
1answer
48 views

Wonder how to evaluate this factorial $\left(-\frac{1}{2}\right)!$

I've learned factorial. But today I saw a question which I don't know how to start with: $$\left(-\frac{1}{2}\right)!$$ Can anyone explain how to solve it? Thanks
1
vote
5answers
71 views

In the sequence $1,3,7,15,31\ldots$ each term is $2\cdot\text{immediately preceding term}+1$. What is the $n$-th term?

I readily see that it is $2^n-1$, but how can I deduce the $n$-th term from the given pattern i.e. $n$-th term $= 2\cdot(n-1)\text{th term} + 1$ without computation.
-4
votes
1answer
75 views

How to find the sum of $\frac{1}{2}+\frac{2}{5}+\frac{4}{9} +\frac{8}{14} \cdots$

Problem : How to find the sum of $\frac{1}{2}+\frac{2}{5}+\frac{4}{9} + \frac{8}{14}\cdots$ Please suggest how to find the sum of such series upto n terms Thanks
0
votes
1answer
15 views

Is there a relation between arithmetic and geometric series for the same a and d=q.

Multiplication is repeated addition, so is there an explicit relation between arithmetic and geometric series if the first term is a and common difference d is equal to the common ratio q. Is it even ...
1
vote
0answers
20 views

Clarify the definition of highest coefficient of a polynomial.

The highest coefficient of a polynomial is given to be 1. Case 1) $ ax^2+bx+c $ Case 2) $ c(x-a)(x-b) $ I am specifically interested in the case 2 polynomial. Edit: I am actually working on a ...
-10
votes
2answers
117 views

How to factor a high-degree polynomial? [on hold]

How to factor this expression $x^6+7x^3+10$ ?
1
vote
1answer
27 views

Finding the domain of a difficult inverse

$f(x)=\frac{3x+5}{-6x+2}$ , largest possible domain Find $f^{-1}(x)$ of this 1-1 function and the domain. So I wrote the equation as $$y=\frac{3x+5}{-6x+2}$$ Interchanged x and y, and made y ...
0
votes
0answers
27 views

Biological modelling math question?

I am trying to write a biological model that models protein interaction. I am having an issue with one aspect. Lets say protein A and protein B interact with eachother to form complex AB. Now every A ...
0
votes
2answers
49 views

Operation with Sigma

How demonstrate that operation: $$1)\sum _{k=1}^{2n}\frac{\left(-1\right)^{k-1}}{k}\:+2\sum _{k=1}^n\left(\frac{1}{2k}\right)\:=\:\sum _{k=1}^{2n}\left(\frac{1}{k}\right)$$ $$2)\sum ...
2
votes
2answers
37 views

Solving for Explicit Form (Differential Equations step)

I have a differential equation $(\alpha)$: $$I'=r\cdot I(S-I) \text{ , }\ I(0)=I_0.$$ Where $r$ is a positive constant. Also, we are given the fact that: $$ \lim_{t\to\infty} ...
0
votes
1answer
18 views

Is it possible to rearrange this term in the form I need?

If I have a term in the following form: $$ 2\frac{(ak + bk - ab)}{(a^2+b^2+k^2)} $$ is it possible to rearrange it into a term like this? $$ 2*f(a, k, b) + f(a, k, b)^2 $$ f can by any type of ...
1
vote
0answers
17 views

$k \in \mathbb{N}$ such that given prime set $X= \{p_1,p_2,,..,p_a\}$, $p_1p_2…p_a - nk$ cannot be decomposed into $a$ prime factors in $X$

How does one find minimum $k \in \mathbb{N}$ such that given prime set $X= \{p_1,p_2,,..,p_a\}$ and $X \subset \mathbb{N}$, $\prod\limits_{i=1}^{a}p_i - nk$ for any $n \in \mathbb{Z}$ cannot be ...
0
votes
2answers
51 views

Let $a_1, \ldots, a_n$ be distinct positive integers. Show that $\sum a_n/n^2\ge\sum 1/n$ [on hold]

Let $a_1, \ldots, a_n$ be distinct positive integers. Show that $$\frac{a_1}{1^2} + \frac{a_2}{2^2} + \cdots + \frac{a_n}{n^2} \geq \frac{1}{1} + \frac{1}{2} + \cdots + \frac{1}{n}.$$
8
votes
11answers
270 views

How to explain to a 14 years old that $\sqrt{(-3)^2}$ isn't $-3$?

$\require{cancel}$ I had this problem yesterday. I tried to explain the kid this: $$\sqrt{(-3)^2}=3,$$ and he immediately said: My teacher told us that we can cancel the square with the square root, ...
0
votes
2answers
46 views

Prove, that x, y, z fulfill this equation

I've got the following problem I can't solve myself. Prove, that if x, y, z fulfill this equation: $$ x+y=z+2015 $$ $$ x^2+y^2=z^2+2015^2 $$ then: $$ x^3+y^3=z^3+2015^3 $$
0
votes
1answer
18 views

Pre-Algebra Fractional Exponent Question

Why does $t^{\frac{3}{2}} \cdot t^{\frac{1}{2}} = t^2$? What I tried to do was multiply the exponents together $\frac{3}{2} \cdot \frac{1}{2} = \frac{3}{4}$ so my final answer was $t^{\frac{3}{4}}$ ...
0
votes
1answer
35 views

Predicting increasing income within a given time

I have this game where player earns "gold" and "reputation". With higher reputation, the player nets more income (gold). Each reputation points yields 1% bonus in income (gold) For every 20 gold ...
1
vote
2answers
31 views

Assistance with polynomial factorization

Is it possible to factorize $x^6 + 4x + 8$? If so what is the fully simplified form, and what would be the steps to get there?
0
votes
3answers
36 views

Solve the Equation.

$$ \begin{bmatrix} 1 & 1 \\ 1 & 1 \\ \end{bmatrix} \begin{Bmatrix} v_1 \\ v_2 \\ \end{Bmatrix}= \begin{Bmatrix} 0 \\ 0 ...
0
votes
1answer
19 views

Determine if a function is even or odd

Let $f:\mathbb{R}\to\mathbb{R}$. Define $h:\mathbb{R}\to\mathbb{R}$ by $$h(x)=f(x)\{f(x)+f(-x)\}$$ Then, which of the following option(s) is/are correct ? (A) h is even for all f (B) h is odd for ...
3
votes
2answers
42 views

How does $y=|x+3|+4$ become $y=\frac{1}{2}|2x+3|+4$ (compositions and translations)

Today, I had a test question that was bothering me because my friend and I had different answers to it. It's a grade 12 math question. It's telling us to explain the changes that were made to the ...
-3
votes
2answers
49 views

Quadratics Homework help [on hold]

The height in feet of an acrobat who jumps from a trampoline $10$ feet in the air to a large mat on the ground can be modelled by the function $$F(x)=-8x^2+16x+10,$$ where $x$ is the time in seconds ...
1
vote
4answers
80 views

Roots to the quartic equation, $(x+1)^2+(x+2)^3+(x+3)^4=2$

Solving with Mathematica gives me the four roots, $$x=-4,-2,\dfrac{-7\pm\sqrt5}{2}$$ Is there some trick to solving this that doesn't involve expanding and/or factoring by grouping?
1
vote
1answer
18 views

Equation for calculating percentile of data point on normal curve

I'm taking pre-calculus right now, and I'm trying to make a statistics program, so I want to calculate the percentile of a score on a normal curve without using a "standard normal table." I can ...
0
votes
1answer
30 views

limt of the function as $\mu\rightarrow\infty$ or $\mu\rightarrow-\infty$ .

$\lim_{\mu\rightarrow\infty}\frac{\exp(\bar x-\mu)^2}{(\bar x-\mu)}=? $ Also, $\lim_{\mu\rightarrow-\infty}\frac{\exp(\bar x-\mu)^2}{(\bar x-\mu)}=? $ I know, ...
0
votes
1answer
18 views

Using given ratios, solve for an unknown quantity using system of equations. [on hold]

Jim bought some chocolates and gave half of them to ken. Ken bought some sweets and gave half to Jim. Jim ate $12$ sweets and Ken ate $18$ chocolates. After that,the number of sweets and chocolates ...
4
votes
1answer
45 views

Is $x(x!)^{1/x}$ an increasing function of $x$, for $x > 0$?

Is $x(x!)^{1/x}$ an increasing function of $x$, for $x > 0$? Here $x!$ is the factorial of $x$. Sure, I do know differential calculus, but my problem is that I do not know how to compute for the ...
0
votes
0answers
24 views

When the company will stop production?

Given total costs function $C(q)=100+10q-6q^2+3q^3$. For which price the company will stop production given that all of the fixed costs are sunk? Do not know how to approach those type of questions, ...
0
votes
2answers
28 views

Proposition from Liber Quadratorum

Prove the following proposition: For any odd square number $x$, there is an even square number $y$, such that $x + y$ is a square number
1
vote
2answers
65 views

How to solve for $x$ in $2(x-5) + 4 (x-3) = -30$

In $2(x-5) + 4 (x-3) = -30$... I'm very confused as to how to solve for $x$, the correct response is $-2$, but I keep getting $4/3$.
1
vote
2answers
47 views

Can I make an assumption about arbitrary numbers in a proof?

Assume $ x_1, x_2... x_{10} $ are different numbers and $ y_1, y_2... y_{10} $ are some arbitrary numbers. Prove that there exists some unique polynomial of degree not exceeding 9 such that: ...
0
votes
2answers
38 views

What is the area of $[r = \frac{4}{2 - \cos \theta}]$?

It makes an ellipse, but I'm unsure where to go from here.