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
1answer
9 views

Existence of a specific invertible matrix

I am an homework question from course in linear algebra, which I don't know how to solve. I need to know if $\exists A_3$, A is a invertible matrix that holds $A^2=-I_3$ , when I is the identity ...
0
votes
2answers
43 views

No. of real roots of $2^x = 1+x^3$

No. of real roots of $2^x = 1+x^3$ $\bf{My\; Try::}$ Let $f(x)=2^x-x^3-1\;,$ Then $f'(x)=2^x\cdot \ln(2)-3x^2$ and $f''(x)=2^{x}\cdot (\ln 2)^2-6x$ and $f'''(x)=2^x\cdot (\ln2)^3-6$ and ...
0
votes
1answer
33 views

Solve that equation [on hold]

Solve equations - $\sqrt{\frac{x+7}{x+1}}+8=2x^2+\sqrt{2x-1}$ $\sqrt{\frac{6}{3-x}}+\sqrt{\frac{8}{2-x}}=6$ $\sqrt[3]{2x+1}-\sqrt[3]{3x^2+x-1}=\sqrt{x}-1$
0
votes
3answers
30 views

Simplify $\frac {3^{(-3+x)}6^{(3-x)}}{3\cdot4^x}$ [on hold]

$$\frac {3^{(-3+x)}6^{(3-x)}}{3\cdot4^x}$$ What is the simplest form?
0
votes
1answer
32 views

Value of indeterminate form — $a_n \to \infty \wedge b_n \to 0$, $\lim_{n\to\infty}a_n\cdot b_n = ?$

$A_n$ and $B_n$ are sequences and $B_n\to 0$ and $A_n\to\infty$. $\lim_{n\to\infty}A_nB_n$ should be equal to $0$ OR $+\infty$ OR $-\infty$? I need to answer yes/no about this problem. I know the ...
0
votes
0answers
9 views

Cuboid with natural number diagonals

I was trying to solve the unsolved problem of finding a cuboid with natural no. sides, face diagonals and space diagonal just as a pastime. I came across the following question. $A=(m-n)(x-y)$ ...
0
votes
5answers
56 views

show that $(1+x^2)(1+x^4)(1+x^8)\cdots (1+x^{2^n}) = \frac{1-x^{2^{(n+1)}}}{1-x^2}$

I am trying to solve the following question in my textbook, one way to go at this would probably be to use induction to prove the statement. But I am looking for alternativ ways to prove this. ...
4
votes
5answers
254 views

Find any polynomial given any 2 points

Is there any way I can find a polynomial given any 2 points (with x coordinate OF MY CHOICE): Let's say there's some polynomial I don't know($p(x)=2x^3+x^2+3$), but my machine will give me an output. ...
2
votes
1answer
36 views

$f(x)$ be a polynomial with integer coefficients and $f(0) = 1989$ and $f(1) = 9891$, then no. of polynomial

Let $f(x)$ be a polynomial with integer coefficients and $f(0) = 1989$ and $f(1) = 9891$. Then prove that $f(x)$ has no integer roots. $\bf{My\; Try::}$ Let $f(x) = ...
0
votes
1answer
19 views

Solving a system of complex equations

$$u = (1+i, i), v = (1-i, 2i), w = (2,3+i)$$ I'm asked to find is there's $z$ such that: $$v = zu$$ So if I suppose $z = a+bi$ I have the system: $$(1-i, 2i) = (a+bi)(1+i, i)\implies\\(1-i, 2i) = ...
0
votes
3answers
25 views

Why does $\frac{1}{4}x^2 + \frac{1}{2} + \frac{1}{4x^2} = (\frac{1}{2}x + \frac{1}{2x})^2$

$\frac{1}{4}x^2 + \frac{1}{2} + \frac{1}{4x^2} = (\frac{1}{2}x + \frac{1}{2x})^2$ This is part of a solution to a more complex problem. Can someone explain what method was used here and how it works? ...
-4
votes
0answers
44 views

Is this function is Invertible? [on hold]

Let f be a function from $\Bbb R\rightarrow\Bbb R$ with $f(x)=x^2 $ $f$ is not one-to-one and $f$ is onto function, is $f$ is invertible or is $f$ onto function?
0
votes
4answers
38 views

Finding the $x$-intercept when variable has fractional exponent

The equation is $$2x-3x^{\frac 23}+4 = 0$$ How would one go about finding the x-intercept(s) of this equation? I have tried, but am unable to isolate the $x$. EDIT: Changed from g(x) = expression ...
2
votes
2answers
70 views

A positive integer $n$ is such that $1-2x+3x^2-4x^3+5x^4-…-2014x^{2013}+nx^{2014}$ has at least one integer solution. Find $n$.

A positive integer $n$ is such that $$1-2x+3x^2-4x^3+5x^4-...-2014x^{2013}+nx^{2014}$$ has at least one integer solution. Find $n$.
0
votes
1answer
49 views

Find consumer demand as a function of time, given the demand equation and price

An importer of Brazilian coffee estimates that local consumers will buy approximately $Q(p)= 4374/p^2$ kg of the coffee per week when the price is $p$ dollars per kg. It is estimated that $t$ weeks ...
2
votes
3answers
32 views

A polynomial $f(x)$ and its behavior as $f(t)>5$

Let $f(x)$ be a polynomial with integer coefficients. Suppose there are four distinct integers $p,q,r,s$ such that $f(p) = f(q) = f(r) = f(s) = 5$. If $t$ is an integer and $f(t)>5$, what is the ...
3
votes
3answers
55 views

Proving $x^2 - y^2 + z^2 \gt (x - y + z)^2$ [on hold]

Prove that $$x^2 - y^2 + z^2 > (x - y + z)^2$$ where: $x < y <z$ for all natural numbers. Thank for help.
6
votes
3answers
41 views

If $f(x) $ and g(x) are functions such that $f(x+y) =f(x)g(y) +g(x) f(y) $ then …

Question : If $f(x) $ and g(x) are functions such that $f(x+y) =f(x)g(y) +g(x) f(y) $ then $\begin{vmatrix} f(\alpha) & g(\alpha) & f(\alpha + \theta) \\ f(\beta) & g(\beta) & f(\beta ...
0
votes
0answers
29 views

Multi-ruled combinatorics problem (need this for my lab)

I need to know this for practical purposes and not homework, learning etc.. Say I have 3 electrodes A,B and C. Say I also have 3 electrolytes A,B and C. If electrode A has to be in electrolyte A, ...
3
votes
1answer
22 views

Function notation meaning: $f: \{a,b\} \to a$ - Zorich - MA I - p18

I have some notation I haven't seen before: $$f: \{a,b\} \to a\text{ and } g:\{a,b\}\to b$$ What does this mean? We are mapping from some $X=\{a,b\}$ to some $Y=a$? So pretty much we are always ...
1
vote
0answers
43 views

Which math class next

I just finished and Algebra for Calculus class this semester. I'm trying to work up to taking calculus (have to do up through calc 3). One person told me I should take trig next, and another calculus. ...
-2
votes
1answer
32 views

How is this problem solved? Steps? And what is the answer?

A company is planning to manufacture portable satellite radio players. The fixed monthly cost will be 300,000 and it will cost $10 to produce each player. a. Write the average cost function, C, of ...
-1
votes
1answer
35 views

How do you solve this word problem? And what is the answer? [on hold]

The data for the system's outflow can be modeled by the formula $$B = 0.07x^2 + 47.4x + 500$$ where $B$ represents the amount paid in benefits, in billions of dollars, $x$ years after 2004. According ...
0
votes
0answers
26 views

What is the point of reflection of this function

$$y = 3x(x+5)^{2/3}$$ Is there some kind of trick to simplify it?
0
votes
2answers
19 views

How can a given length of something yield different sum in square meters?

How can a rope of say 100 meters yield different return in square meters, based on how you divide each side? E.g. 10m x 10m = 100m2 15m x 5m = 75m2 Now of course I see that based on how you choose ...
1
vote
2answers
33 views

Math question related to Train

Two trains X and Y, leaves from point A and B towards B and A at the same time, after meeting each other they takes 4 hr 48 min and 3 hr 20 min to reach the point B and A. If the speed of train X is ...
1
vote
1answer
27 views

Graphing inequalities on a number line

What software or websites for graphing inequalities on a real number line?
-6
votes
3answers
72 views

Is it possible to find [on hold]

If $$\frac {(a-b)(c-a)}{(b-c)(d-c)}=\frac {2012}{2013}$$ then find the value of $\dfrac {(a-c)(b-d)}{(a-b)(c-d)}$ in terms of numbers Note: $a,b,c,d$ are integers
-1
votes
1answer
48 views

An easy question regarding Algebra [on hold]

Three schools $A, B, C$ have a total of $480$ students. Ten per cent of the students of school $A$ are going camping and the percentages for school $B$ and school $C$ are 8.5% and 15% respectively. ...
1
vote
2answers
31 views

how to prove $a+b-ab \le 1$ if $a,b \in [0,1]$?

Given: $0 \le a \le 1$ $0 \le b \le 1$ Prove: $a + b - ab \le 1$
3
votes
3answers
52 views

Factorising quadratics - coefficient of $x^2$ is greater than $1$

In factoring quadratics where the coefficient of $x^2$ is greater than $1$, I use the grouping method where we multiply the coefficient and constant together and then factor. My question is can ...
0
votes
1answer
74 views

Questions about $f(n)=3+\frac{12}n$

Experimental Psychology: To study the rate at which animals learn, a psychology student performed an experiment in which a rat was sent repeatedly through a laboratory maze. Suppose the time in ...
6
votes
2answers
352 views

Quadratics with roots as integers; possible values of a

Suppose $a$, $b$ are real numbers such that $a+b=12$ and both roots of the equation $x^2+ax+b=0$ are integers. Determine all possible values of $a$. I don't know how to go about doing this without ...
1
vote
1answer
23 views

Remainders and polynomial division

Completley impromptu, one of my extended middle school students asked a question about her additional maths she was studying outside of school. For a certain polynomial, f(x), the remainder on ...
0
votes
1answer
29 views

Solving for the roots of a polynomial

Suppose we have a polynomial of the form: $$-x^3+3x^2+9x-27=0$$ Is there an easy way to find the solutions of $x$? I know that they will be factors of $27$, so I begin by factoring $27$ into ...
1
vote
2answers
25 views

Solving equations with exponentials and trig algebraically

Is it possible to algebraically solve an equation of the following form? $A\sin(x)+Be^x=C$ If so, how?
0
votes
2answers
43 views

Faster way to for $z^3 = -2 (1+i \sqrt 3) \bar z$ than complex algebra

What is the fastest way to solve for $z^3 = -2 (1+i \sqrt 3) \bar z$? I know how to do this using complex algebra. but that takes a long time. Can someone show me a faster way?
2
votes
2answers
39 views

Simplifying $\frac{\sqrt{3}}{2\sqrt{3}+1} + \frac{\sqrt{3}}{11}$

I realize that is a basic math problem, but I am still having problems with it. The expression $$\frac{\sqrt{3}}{2\sqrt{3}+1} + \frac{\sqrt{3}}{11}$$ equals one of the following: $2\sqrt{3}-1$ ...
-1
votes
0answers
26 views

Let $X = \mathbb{R}$ and $Y = \left \{ x \in \mathbb{R}\mid x ≥ 1 \right \}$. Define $G : X → Y$ by $G(x) = e^{x^2}$. Prove that $G$ is onto. [duplicate]

Let $X = \mathbb{R}$ and $Y = \left \{ x \in \mathbb{R}\mid x ≥ 1 \right \}$. Define $G : X → Y$ by $G(x) = e^{x^2}$. Prove that $G$ is onto.
5
votes
1answer
60 views

no. of real solution of the equation $1+8^x+27^x = 2^x+12^x+9^x.$

The no. of real solution of the equation $1+8^x+27^x = 2^x+12^x+9^x.$ $\bf{My\; Try::}$ Let $2^x=a>0$ and $3^x=b>0\;,$ where $x\in \mathbb{R}$ So equation convert into $1+a^3+b^3 = a+a^2b+b^2$ ...
0
votes
2answers
30 views

Rearrange $y = xa-zc$ so that $a-c$ is on one side of the equation.

Is it possible to rearrange the following equation so that $a - c$ is on one side of the equation? $$ y = xa-zc $$ Thanks!
0
votes
1answer
7 views

Is every zonal homogeneous polynomial a polynomial on the unit sphere?

Let $$P_k(x_1\ldots x_n)=\sum_{\lvert \alpha\rvert=k} c_\alpha x_1^{\alpha_1}\ldots x_n^{\alpha_n}, \qquad (x_1\ldots x_n)\in \mathbb{R}^n$$ be a homogeneous polynomial of degree $k$. Assume that ...
2
votes
4answers
69 views

Why $(x-5)^2-4$ can be factorised as $(x-5-2)(x-5+2)$

I would like to understand why $(x-5)^2-4$ can be factorised as $(x-5-2)(x-5+2)$ I am particularly concerned with the term, $-4$.
-1
votes
2answers
24 views

Solving Inequalities with the use of their properties and cases [on hold]

Solve following inequality $$\dfrac4x + 3 \gt \dfrac2x + 1$$ and then graph the solution set on real number line.
5
votes
4answers
89 views

Prove that $13\vert(3^{n+1} +3^{n} +3^{n-1})$

Prove that $3^{n+1} +3^{n} +3^{n-1}$ is divisible by $13$ for all positive integral values of $n$
0
votes
2answers
23 views

Using the basic laws of exponent [on hold]

I have some problems with this question. Please help me. Thanks Simplify given expression$$ a^2 (abc)^{-2} a^3 b^7 $$ What are exponents of $a$, $b$, and $c$? I get $3,5,-2$ as exponents of ...
0
votes
2answers
18 views

Express the given expression as a single logarithm

Express $$2 \ln (2 - x) + 3 \ln (x^2 - 5)$$ as a single logarithm. Can anyone help me with this question? Thanks
1
vote
3answers
45 views

Show that $2(a^3+b^3+c^3)>a^2(b+c)+b^2(c+a)+c^2(a+b)>6abc$

If $a,b,c$ are positive real numbers, not all equal, then prove that $$2(a^3+b^3+c^3)>a^2(b+c)+b^2(c+a)+c^2(a+b)>6abc$$ How can I show this?
7
votes
3answers
70 views

valid proof of series $\sum \limits_{v=1}^n v$

$$\sum \limits_{v=1}^n v=\frac{n^2+n}{2}$$ please don't downvote if this proof is stupid, it is my first proof, and i am only in grade 5, so i haven't a teacher for any of this 'big sums' proof: if ...
0
votes
1answer
50 views

Find exact value of $\sin\left(\dfrac x2\right) $

I have tried this problem over and over but can not get it. Can anyone provide a solution? Given $\sin(x) = -\dfrac67$ and $\tan(x)\gt0$ , find the exact value of $\sin\left(\dfrac x2\right) $.