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
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0answers
10 views

The irreducibility of $a^{4n}+b^{4n}$

How to prove that $a^{4n}+b^{4n} $, for any natural number $ n $, is irreducible over the rationals?
0
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3answers
25 views

Continous function

There are two functions g(x)= ($(2x+1)^{1/2}$-$1$)/x , where x is not equal to zero = 1 , x=0 h (x) = $x^9 - 6x^8 -2x^7 + 12x^6 +x^4 -7x^3 + 6x^2 + x-7$ ...
1
vote
2answers
26 views

build absolute value equations know solution

We have absolute value equations with unknown coefficients: $$|x + a| = b$$ and we know the solutions: $$x = 11 \text{ and } x = 5$$ We need to find $a$ and $b$. From $$11 + a = b \\ 5 + a = -b$$ we ...
-1
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2answers
37 views

Getting $\sin^2$ and $\cos^2$ values from $\sin^2 \alpha = \frac{1}{4} \cos^2 \alpha = \frac{1}{4}(1 - \sin^2 \alpha)$

How can you get from $$\sin^2 \alpha = \frac{1}{4} \cos^2 \alpha = \frac{1}{4}(1 - \sin^2 \alpha)$$ to $$\sin^2 \alpha = \frac{1}{5} \\or \\ \cos^2 \alpha = \frac{4}{5}.$$ Sorry, but I can't see ...
-3
votes
1answer
34 views

Problem about water and time

You can fill a container with water from a hot water tap in $80$ minutes. The container can be filled with water from a cold water tap in $48$ minutes. How long does it take to fill the tub, if you ...
5
votes
3answers
47 views

Factorization of $x^5+x$

I need to make the decomposition in $\mathbb{R}$ of: $$x^5+x$$ Here my steps: $$x(x^4+1)$$ $$x\big[ (x^2)^2+1 \big]$$ $$x\big[(x^2+1)^2-2x^2\big]$$ how should I proceed?
1
vote
1answer
29 views

Evaluating the arc length integral $\int\sqrt{1+\frac{x^4-8x^2+16}{16x^2}} dx$

Find length of the arc from $2$ to $8$ of $$y = \frac18(x^2-8 \ln x)$$ First I find the derivative, which is equal to $$\frac{x^2-4}{4x} .$$ Plug it into the arc length formula ...
1
vote
0answers
9 views

Number of terms in multivariate polynomial

We know that the number of terms in a univariate polynomial of degree n is n+1. But what about if there are multiple variables: for eg: for variables $x,y$ polynomial of degree 2 will have: ...
4
votes
0answers
22 views
+100

Maximum value of the smallest number of operations to obtain configuration from original configuration

Let $n$ be a positive integer. There are $n(n+1)/2$ marks, each with a black side and a white side, arranged into an equilateral triangle, with the biggest row containing $n$ marks. Initially, each ...
0
votes
0answers
25 views

Solving an inequality systematically

A question states: "Find all $n >3$ such that $$ \frac{1}{n^{1.1}}<\frac{1}{n \ln n}" $$ Here's my step: $$ n^{1.1}>n \ln n $$ $$ n^.1>\ln n $$ $$ n >(\ln n)^{10} $$ Setting $(\ln ...
0
votes
1answer
15 views

Show inequality is correct by simplification

This may be a bit of a silly question, but I was wondering if there is any way to show with some simplification (without using a calculator) that clearly $32-16\sqrt2 < 8\sqrt2$ (since it ...
1
vote
1answer
11 views

simplification question on Diff. Eq. Solution

Can anyone explain this simplification from y^-2 to y? If you distribute the x^4 through, you obtain y^-2 = (2/x) + Cx^4. This leads to y^2 will equal the reciprocal of what I just wrote. Where in ...
0
votes
1answer
23 views

Parametric Problem

i have a question on parametric.. The question states A vector equation $(x,y) = (2,-1) + t(3,2)$. Write as a parametric equation. Show a table with x,y values. Sketch a picture of vector ...
0
votes
3answers
31 views

Let $z \in C^*$ such that $|z^3+\frac{1}{z^3}|\leq 2$ Prove that $|z+\frac{1}{z}|\leq 2$

Problem : Let $z \in C^*$ such that $|z^3+\frac{1}{z^3}|\leq 2$ Prove that $|z+\frac{1}{z}|\leq 2$ My approach : Since : $(a^3+b^3)=(a+b)^3-3ab(a+b)$ $\Rightarrow ...
0
votes
1answer
23 views

Algebra question - Formulas

Really having a hard time with this question. any help would be appreciated. Suppose that a cliff diver's height (in feet) after t seconds is given by the model $H = −16t^2+48t+28$ . Find the height ...
3
votes
4answers
42 views

how to verify $\frac{\sin(x)\cos(x)}{\cos^2(x)-\sin^2(x)}=\frac{\tan(x)}{1-\tan^2(x))}$? [on hold]

How would I verifty the following trig identity? $$ \frac{\sin(x)\cos(x)}{\cos^2(x)-\sin^2(x)}=\frac{\tan(x)}{1-\tan^2(x)} $$ I am not sure how to start.
1
vote
1answer
539 views

Transformation matrix from a translated-rotated coordinate system to the general coordinate system

In Figure 1, suppose $XYZ$ (in black) as my general coordinate system and $X'Y'Z'$ (orange) as another system with parallel axes respect to $XYZ$. Consider $xyz$ (green) is my 3rd coordinate system ...
0
votes
3answers
28 views

Find particular solution to nonhomogeneous DE $y'+y=x^2+\sin{x}+\cos{x}$

I'm new to nonhomogeneous DE's and I have come across this DE: $$y'+y=x^2+\sin{x}+\cos{x}$$ which I'm supposed to provide a general solution to. However, I get stuck with the particular solution. The ...
5
votes
1answer
85 views

Why does this approximation work (and why does it fail)?

I have a function $$f(x)=\frac{e^{-x}}{x}$$ and I am trying to find an expression for the inverse function $f^{-1}(x)$. So far I have come up with the approximation: $$\hat{f}^{-1}(x)=\left( ...
0
votes
0answers
21 views

Proving that if $m,n,p,q\in\mathbb{Z^+}, \sqrt[p]{m}\in\mathbb{R}\setminus\mathbb{Q}$ then $\sqrt[p]{m}+\sqrt[q]{n}\in\mathbb{R}\setminus\mathbb{Q}$

If $\sqrt{m}+\sqrt[q]{n}=r$ rational, the rationality of $\sqrt{m}$ is derived expanding $(r-\sqrt{m})^q$ using the binomial theorem: after rearrangement, isolating the terms containing odd powers of ...
1
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1answer
28 views

Let a,b,c be distinct non zero complex numbers with $|a|=|b|=|c|$ If each of …

Problem : Let a,b,c be distinct non zero complex numbers with $|a|=|b|=|c|$ If each of the equations $az^2+bz+c=0$ and $bz^2+cz+a=0$ has a root having modulus 1, then prove that : ...
0
votes
1answer
17 views

Let a be a positive real number and let $M_a=\{z \in C^* : |z+\frac{1}{z}|=a\}$ Find the minimum… [duplicate]

Problem : Let a be a positive real number and let $M_a=\{z \in C^* : |z+\frac{1}{z}|=a\}$ Find the minimum and maximum value of $|z|$ when $z \in M_a$ My approach : $|z+\frac{1}{z}|=a$ Squaring ...
3
votes
1answer
44 views
+50

Jensen-like averaging inequality on integers

Let $\mathbb{Z}^*=\mathbb{Z}^+\cup\{0\}$. Let $f:\mathbb{Z}^*\rightarrow\mathbb{R}$ be a nondecreasing function such that $f(a+b)\leq f(a)+f(b)$ for all $a,b\in\mathbb{Z}^*$. Is it true that for all ...
1
vote
3answers
48 views

Solve the equation with radicals: $x^2- x \sqrt[4]{2} (1+ \sqrt[4]{2} ) + \sqrt[4]{8} = 0$

Solve this equation : $ x^2- x \sqrt[4]{2} (1+ \sqrt[4]{2} ) + \sqrt[4]{8} = 0 $ I've calculated $ \Delta $ and it is something like this: $ \sqrt[4]{4} + 4 \sqrt[4]{2} + 4 - 4 \sqrt[4]{8} $ I ...
1
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2answers
27 views

Proceed With Radical Equations $\sqrt{2x-8} + 8 = x$

I have a question about how to solve the following equation: $$ \sqrt{2x-8} + 8 = x $$ It doesn't seem so difficult as we can just isolate the radical and be left with $\sqrt{2x-8} = x - 8$. further ...
7
votes
4answers
368 views

Find the value of $\frac{a^2}{a^4+a^2+1}$ if $\frac{a}{a^2+a+1}=\frac{1}{6}$

Is there an easy to solve the problem? The way I did it is to find the value of $a$ from the second expression and then use it to find the value of the first expression. I believe there must be an ...
0
votes
2answers
32 views

Inverse function on interval where it is one-to-one

I have to find inverse function of ${\displaystyle{f(x)=\frac{\log x}{\log 2}+\frac{\log 2}{\log x}.}}$ ${\displaystyle{y=\frac{\log x}{\log 2}+\frac{\log 2}{\log x}}}$ ...
0
votes
1answer
57 views

Riemann Sum of $f(x)=2^x$

Using Riemann Sums, how can I compute the integral $$\int_{0}^{2} 2^x dx$$ I don't know how can I take the Partition and then compute the sums , someone can help to understand this method of Riemann ...
6
votes
3answers
96 views

Is there an infinite sequence of real numbers $a_1, a_2, a_3,… $ such that ${a_1}^m+{a_2}^m+a_3^m+…=m$ for every positive integer $m$?

Is there an infinite sequence of real numbers $a_1, a_2, a_3,...$ such that ${a_1}^m+{a_2}^m+a_3^m+...=m$ for every positive integer $m$? I tried assuming that the sequence $a_1^m, a_2^m,...$ ...
2
votes
1answer
26 views

Maximum of $(ab+cd)(ac+bd)(ad+bc)$

Let $a,b,c,d\ge 0$ satisfy $a+b+c+d=4$. Find the maximum value of $(ab+cd)(ac+bd)(ad+bc)$. When all of the variables are $1$, the value is $8$. Using the AM-GM inequality gives ...
1
vote
2answers
18 views

Mean Value Inequalities

I am working through an Algebra book, and am having trouble understanding exactly how $$ q(x^p-1) - p(x^q-1) >< 0 $$ is equivalent to $$ (x-1)[q(x^{p-1} + x^{p-2} + ... + 1)-p(x^{q-1} + ...
4
votes
2answers
176 views

Find the domain

I have been a bit confused about finding the domain of these functions. 1) $\dfrac{12}{2x+3}$ 2) $\dfrac{4x-3}{x^2-81}$ 3) $\dfrac{x^2 -3x -18}{x-6}$ So I solved for $x$ and then those were the ...
0
votes
1answer
39 views

Calculator Shortcuts for Derivative?

I have a TI-84 and I am in a business calculus class. Our teacher doesn't give us much time on tests and I am a bit slow so I was wondering if there was any way to do some of these problems on the ...
0
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2answers
33 views

log to exponential form, but with number in front of log

So I understand how to put a log equation into exponential form. For example, $y = \log_2(x)$ is $2^y = x.$ However, I don't understand what to do when there is a number in front of $\log$, such as ...
0
votes
8answers
91 views

Least value of $a$ for which $4ax^2 + \frac{1}{x} \geq 1$

Find the least value of $a \in R$ for which $4ax^2 + \frac{1}{x} \geq 1$, for all $x>0$. The equation will transform into (Using $x>0$) $4ax^3-x+1\geq 0$ But I don't know how to deal with ...
0
votes
0answers
26 views

solving equation invloving both algebraic and trigonometric terms

$$ x\sin(3)+3\sin(x)-xlog(3)+3log(x)=10$$ I need to know a method that finds all the possible values of x that satisfy the above equation.
0
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3answers
3k views

SHL numerical reasoning question

Hi I've screen shot the question and attached it here. I have tried numerous methods to solve it but can't seem to get an answer which correlates to one of the choices given. So i need help on working ...
1
vote
2answers
24 views

sum of all positive integral values of $a\;,$ for which equation $\lfloor x \rfloor ^3+x-a=0$ has solution

The sum of all positive integral values of $a\;,$ Where $a\in \left[1,1500\right]$ for which the equation $\lfloor x \rfloor ^3+x-a=0$ has solution, Where $\lfloor x \rfloor $ Represent floor ...
0
votes
2answers
21 views

Prove that $aRb$ if $a = 2^kb$ is an equivalence relation.

Let $R$ be a relation on the set of integers given by $aRb$ if $a = 2^kb$, for some integer $k$. show that $R$ is an equivalence relation. I don't understand how it will be equivalence. Is it the ...
2
votes
3answers
62 views

How to solve the equation $(2x + 1)(2x + 3) = 143$ without using the Quadratic Formula?

I have been a bit stuck on this question. The product of two consecutive odd numbers is $143$. Find the next numbers. I have made this into: $$ (2x+1)(2x+3)=143. $$ I got $x_1 = -7$ and $x_2 = -5$ ...
1
vote
2answers
62 views

$8^a=3$ and $3^b=5$ and $10^c=5$ then find $c$ in terms of $a$ and $b$.

if $8^a=3$ and $3^b=5$ and $10^c=5$ then find $c$ using $a$ and $b$. My Attempt: if $8^a=3$ and $3^b=5$ then we can say that $8^{ab}=5$ and then we have $2^{3ab}=10^c$ but i cant solve this ...
75
votes
15answers
5k views

math fallacy problem: $-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1$?

I know there is something wrong with this but I don't know where. It's some kind of a math fallacy and it is driving me crazy. Here it is: $$-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1?$$
0
votes
1answer
13 views

Is there a analytic algorithm to solve for a partially specified constant present in function & its derivative with the rate of change at an x value?

I have a Calculus problem that I am not entirely happy with how I solved it. Given the following information: $$y = x^{k} + x^{k-2}$$ $$(k \in \mathbb{N}) \wedge (k \mod{2} \neq 0) \wedge (k > ...
2
votes
2answers
42 views

Quadratic formula errors

I'm clearly making a silly mistake here, but I can't see it. EDIT: I missed brackets when typing out the expression to calculate. Apologies for timewasting. I have the equation $(2x + 3)(5x + 1)=0$. ...
0
votes
1answer
29 views

Kernel of a polynomial with matrix, $ker(p(A))$

Let $A\in Mat(3,3,\mathbb R)$ a matrix and $\chi_A(x)=p_1(x)\cdot p_2(x)$ the characteristic polynomial. Evaluate $ker(p_1(A))$.$$A=\begin{pmatrix} 0 & 0 & 2 \\ 1 & 0 & 1\\ 0 & ...
0
votes
3answers
62 views

Multiplication of real and complex radicals

If I have, for example, the product $\sqrt{7+\sqrt{22}}\sqrt[3]{38+i\sqrt{6}} $ Can I perform the multiplication or this cannot be done and only remains to leave the product in this form?
1
vote
7answers
86 views

Solve the following equation : $\log_2(x)*\log_4(x)*\log_8(x)=4.5$

I have the following equation : $$\log_2(x)*\log_4(x)*\log_8(x)=4.5$$ Usually, I do post what I made to do, but in this case a friend of mine tackle me with this question after I didn't mess with ...
0
votes
0answers
26 views

Number of solutions of a difference-of-two-squares congruence with prime moduli

Problem: Show that if $p$ is an odd prime then $p-1$ number of ordered pairs $x, y$(unique modulo p) satisfy $x^2-y^2 \equiv a\mod p$ (for some given $a$ coprime to p). When $a \equiv 0 \mod p$ then ...
0
votes
1answer
37 views

$1-x^2e^{-2x-y}+e^{-2x-y}\overset{?}=1-x^2e^{-2x-y}$ [on hold]

I'am looking at a solution of a problema that has this, it it correct? If so why? I'am looking at this and can't find out what step was made to simplify like that. ...
0
votes
1answer
67 views

Create an equation for a description of a rational function

A graph has a $y$-intercept at $-5$, no $x$-intercepts, and discontinuous points at $(-1,-5)$ and $(3, -5)$. I want to form an equation for this graph, but I don't know how the $y$-intercept relates ...