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.

learn more… | top users | synonyms (2)

0
votes
2answers
21 views

Coefficient of product of polynomials.

Suppose we have the polynomial $f(x)$ and another polynomial $g(x)$. How can I find the coefficient of say $x^n$ in the product of the polynomials without actually multiplying. I am not that ...
1
vote
1answer
35 views

Where i am going wrong in solving the inequality?

If $\cos x \left(\cos x+\frac12\right) >0$ then where should $x$ lie in the interval $(0,\pi)$ What I tried When i made two cases i got correct answer but when i used wavy-curve method. I am not ...
0
votes
3answers
22 views

Find the Capacity of the Water Tank?

A water tank has three taps attached, $A,B$ and $D$. $A$ and $B$ fill the water tank completely in $\displaystyle\frac{25}{3}$ minutes and $\displaystyle\frac{25}{2}$ minutes, respectively. ...
0
votes
3answers
44 views

Write $f(x) = x \cdot |x|$ as a piecewise function

$$f(x) = x\cdot|x|$$ I was wondering how this function should look if I expanded it to have the format of a piecewise defined function? I know how to write a piecewise defined function, but the ...
-3
votes
4answers
64 views

Prove that, if $a>c$ and $b>d$, thus $ab>cd$ [closed]

I would like to ask you a question: how could I prove that, if $a>c$ and $b>d$, thus $ab>cd$? Thank you for help. P.s. I forgot to tell you that $a>0, b>0, c>0, d>0.$
1
vote
5answers
121 views

Show that $n^2+11n+2$ is not divisible by $113^2$ ( n is integer)

Show that $n^2+11n+2$ is not divisible by $113^2$ ( n is integer) It's obvious that if we show $113$ doesn't divide $n^2+11n+2$ we are done...
-1
votes
3answers
71 views

Let $(\sqrt{3} + \sqrt{2})^5 = a\sqrt{3} + b\sqrt{2}, a,b \in \mathbb Z$ Find $a+b$.

Let $$(\sqrt{3} + \sqrt{2})^{\color{red}{5}} = a\sqrt{3} + b\sqrt{2}, a,b \in \mathbb Z$$ Find $a+b$. I don't know if that's supposed to be $\color{red}{5}$ or $\color{red}{3}$. By binomial ...
-2
votes
2answers
38 views

Why does $-(3e^{-x})(1-x)-(3e^{-x}) = (-3e^{-x})(2-x)$?

I am looking at an old exam. The first part of the task wants you to differentiate $$ f(x) = 3xe^{-x}, $$ which is $$ f'(x) = 3e^{-x}(1-x) $$ but then, it wants you to differentitate $f'(x)$. While ...
0
votes
1answer
50 views

Is it possible that $2-2\cos^2x$ is equivalent to $1-(2\cos^2x-1)$

There is this exercise and for the first time in my life, I don't want to go to see the solution. Instead, I'm more asking of a tiny help to see if I'm right in my conclusion Kids are getting ...
0
votes
1answer
86 views

About the solution to “Finding the range of $y= \sqrt x + \sqrt{3-x}”$

I was reading the solution of "Find the range of $y = \sqrt{x} + \sqrt{3 -x}$" and I had some points of confusion about the solution posted in the OP. I wrote here my interpretation of the solution. ...
1
vote
0answers
28 views

About the identity $(\vec{R}+\vec{a})\cdot(\vec{R}-\vec{a})=R^2-a^2$

From basic vector algebra, we know that $(\vec{R}+\vec{a})\cdot(\vec{R}-\vec{a})= R^2-\vec{R}\cdot\vec{a}+\vec{a}\cdot\vec{R}-a^2=R^2-a^2$, where the result is independent of the angle between ...
3
votes
3answers
78 views

Solution to $\sqrt{x-2} = 3- 2\sqrt{ x}$

The above question is from Serge Lang's basic mathematics. The question asks if there are any values of x which satisfy the above equation. Serge Lang's answer key states that there is no solution. ...
0
votes
1answer
67 views

can someone explain this simplification for me?? [closed]

Can someone tell me how $$−56−173\,\ln(11)+366\,\ln(13)−\left(\frac{105}2+20\,\ln(2)+366\,\ln(3)\right)$$ simplifies to $$\frac{-217}2−20\,\ln(2)−173\,\ln(11)+732\,{\rm arctanh}\left(\frac58\right)?$$ ...
0
votes
2answers
61 views

Hello i am confused with this quadratic question.

Why is it that when we look at equation for example $(x-2)^2+5$ and the question states "state the minimum point" the minimum point is $5$. I get that the coordinates of minimum point is $(2,5)$ but ...
0
votes
1answer
26 views

solve system inequalities derived from a function

I have this system of inequalities $$ \begin{cases} y^2-3 \geq 0\\ 16y^4-96y^2 \geq 0 \end{cases} $$ the solution for the first inequality is $y\leq -\sqrt{3}$ or $y\geq \sqrt{3}$ and the solution ...
1
vote
2answers
26 views

find the result of $16y^4-96y^2 \leq 0$

$16y^4-96y^2 \leq 0$ I have not clear the last step of this inequality to get the result $-\sqrt{6} \leq y \leq \sqrt{6}$. Change $t=y^2$ and $t^2 = y^4$ $16t^2-96t \leq 0$ I compute the ...
2
votes
1answer
36 views

When you divide the polynomial $A(x)$ by $(x-1)(x+2)$, what remainder will you end up with?

When you divide the polynomial $A(x)$ by $x-1$, you get a remainder of $10$. When you divide $A(x)$ by $x+2$ you get remainder $0$. When you divide $A(x)$ by $(x-1)(x+2)$ what remainder will you end ...
-2
votes
2answers
39 views

How to rearrange this equation [closed]

Can you please help me to solve the following equation for $s$? $$[w-s]^{-0.5}=[(1+r)s]^{0.5}$$
0
votes
2answers
37 views

Ascertaining a from logarithmic equations

I've just been accepted on to a PHD program at Melbourne, studying chemical engineering. I'm working my way through some standard pure and further mathematics books just to get the concepts into my ...
1
vote
1answer
80 views

Solve the following equation for $x$

$(1)$ Solve for $x: \sum_{i=1}^p\frac{1}{(x-x_i)^2}=\sum_{i=p+1}^n\frac{1}{(x-x_i)^2}$ where $x_i$'s are fixed real numbers. Note: It's a generalization of the usual problems like solving ...
2
votes
3answers
46 views

For real numbers $a,b,c$ calculate the value of: $\frac{c}{a+b}+\frac{a}{b+c}+\frac{b}{c+a}$ if we have…

For real numbers $a,b,c$ we have: $a+b+c=11$ and $\frac1{a+b}+\frac1{b+c}+\frac1{c+a}=\frac{13}{17}$, calculate the value of: $\frac{c}{a+b}+\frac{a}{b+c}+\frac{b}{c+a}$ I think we should use a trick ...
2
votes
1answer
35 views

$\lfloor\frac{18}{35}\rfloor+\lfloor \frac{18(2)}{35}\rfloor+\lfloor \frac{18(3)}{35}\rfloor+…+\lfloor \frac{18(34)}{35}\rfloor$

Value of the expression $$\bigg\lfloor \frac{18}{35}\bigg\rfloor+\bigg\lfloor \frac{18(2)}{35}\bigg\rfloor+\bigg\lfloor \frac{18(3)}{35}\bigg\rfloor+....+\bigg\lfloor ...
-1
votes
2answers
35 views

$100 + [110/(1+r)] = [1/ (1+r)] + [(232 /(1+r)^2 ]$

Need to learn how to solve this: $100 + \frac{110}{1 + r} = \frac{1}{1 + r} + \frac{232}{(1 + r)^{2}}$. Checked this site got to the 3rd line and am completely lost. Can someone help me solve for r ...
0
votes
2answers
36 views

Negative Ratio----Math

I have always studied the ratios of the type $a:b$, where a and b are natural numbers and I can also understand the ratio where both a and b are negative. But what I don't get is that when a is ...
2
votes
2answers
79 views

Number of polynomials which are divisible by $x+1$

Let $a,b,c,d$ be four integers (not necessarily distinct) in the set ${1,2,3,4,5}$ . The number of polynomials $f(x)=x^4+ax^3+bx^2+cx+d$ which are divisible by $x+1$ are: $(A)$ Between 55 and 65 ...
0
votes
0answers
40 views

Suppose ($ a_{1} ,…, a_{n} $) is an arithmetic sequence.

Suppose ($ a_{1} ,..., a_{n} $) is an arithmetic sequence. Then: $$ \frac{1}{ a_{1} } +....+ \frac{1}{a_{n} }=؟ $$ Is it possible to obtain high regard.
1
vote
1answer
62 views

Calculating $\sqrt[3]{\sqrt 5 +2}-\sqrt[3]{\sqrt 5 -2}$

We want only the real 3rd root. By calculation, $[\sqrt[3]{\sqrt 5 +2}-\sqrt[3]{\sqrt 5 -2}]^3= 4-3[\sqrt[3]{\sqrt 5 +2}-\sqrt[3]{\sqrt 5 -2}]$ Therefore, the answer is a root of $t^3=4-3t$ , which ...
1
vote
4answers
30 views

On proving the surjectivity of non-injective functions.

As given in my lectures and several other areas, the definition of a surjective function is "a function f from a set X to a set Y is surjective (or onto), or a surjection, if every element y in Y has ...
0
votes
1answer
56 views

What is the square root of a square [duplicate]

What is the $\sqrt{x^2}$? Is it just $x$ or $\pm x$? When is it $\pm x$? So if I had $\sqrt {\sin^2(x)}$ is it then $\sin(x)$? I feel like its just $x$ as $x^2$ is always positive. Thoughts? Thanks
0
votes
1answer
18 views

Total number of integral solutions for the given second degree equation!

First, the problem statement : "Consider the equation $x^{2}+y^{2}-3z^{2}-3t^{2}=0$. The total number of integral solutions of this equation in the range of the first 10000 numbers, i.e., $1\leq ...
1
vote
1answer
30 views

sum of all non real roots of the equation in a bi-quadratic equation

Consider the equation $8x^4-16x^3+16x^2-8x+a=0\;\left(a\in \mathbb{R}\right)\;,$ Then the sum of all non real roots of the equation can be $\bf{OPTIONS::}\;\; (a)\;\; 1\;\;\;\;\;\; (b)\;\; ...
5
votes
3answers
80 views

Proof that the Period of $\sin(x)$ is $2\pi$.

As I was walking through campus today, I had an interesting question pop into my head: How can we prove that the period of $\tan(x)$ is $\pi$ rather than $2\pi$? The answer to this was extremely ...
-5
votes
5answers
54 views

What is the relationship between Onto and One-to-One? [closed]

What is the relationship between one-to-one and onto? Can a one-to-one function be onto? Can an onto function be one-to-one? Must a one-to-one function be onto? Must an onto function be 1-1?
-3
votes
2answers
42 views

How do I solve this inequality? [closed]

How do I solve this equation? $7(-7b-2)<231$ I do not know where the $231$ should move to.
0
votes
2answers
48 views

Math formula to calculate game score

I am developing a mobile game, however I am a little stuck on the part where I need to calculate the players score. I need to come up with a formula to calculate that, and I need your help! Here are ...
1
vote
1answer
33 views

Is the fundamental theorem of algebra valid with polynomial terms like $\bar{z}$ and $\Re (z)$?

If we have a polynomial-equation with complex coefficients (of finite degree), like $$z^3+5z+22=0$$ where $z\in\mathbb{C}$, then we're guaranteed as many complex solutions as the degree of the ...
0
votes
1answer
57 views

General formula for a diagonal parabola.

What is the general formula for a diagonal parabola facing a given direction with a given vertex (x,y)?
3
votes
2answers
47 views

Cyclic System of quadratic equations

Find all solutions to system of the equations $$ \begin{align*} x^2&=a+y\\ y^2&=a+z\\ z^2&=a+x\\ \end{align*} $$ I have only found 2 solutions by setting $x=y=z$ but there can be a total ...
1
vote
3answers
61 views

Why is the ratio of the slope of a line always equal?

Imagine you have one triangle and the hypotenuse is a line $AB$. Now the slope of this triangle is defined as: $$ m =\frac{y_i-y_1}{x_i-x_1} $$ $i$ = any number on the line $AB$ Now instead if ...
0
votes
2answers
36 views

Help with a factorisation problem. Completely factor $4(a^2)(c^2) - (a^2 - b^2 + c^2)^2$

The questions asks to completely factor the following polynomial: $4a^2c^2 - (a^2 - b^2 + c^2)^2$ The closest I can get to completely factoring it is (and I am not sure if this is even correct): ...
8
votes
2answers
2k views

Why do these “equal” logarithms give different answers

This came across a discussion amongst Algebra 2 teachers at my school. We know $a\log x= \log x^a$ Say $2\log x=5$ $\log x^2 =5$ When $\log x=\log_{10} x$ Solving the first equation yields ...
0
votes
2answers
39 views

Find the least value of $k$

The function is as follows: $f(x) = 2x^2 - 6x + 8$ and the domain is less than and equal to $k$. Find the least value of $k$ for which $f(x)$ is one to one. I've completed the square to equal $2(x- ...
1
vote
0answers
33 views

Evaluate $\int \frac{\sec x \:dx}{\sin (2x+\theta)+\sin \theta}$

Evaluate $$I=\int \frac{\sec x \:dx}{\sin (2x+\theta)+\sin \theta}$$ I used $$\sin A+\sin B=2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)$$ we get $$I=\frac{1}{2} \times \int ...
1
vote
1answer
27 views

Do not understand algebra technique used to computer summation

I am going through a practice exam for my Discrete Mathematics class and do not understand the algebra used in the following summation computation. Summation to compute: Answer: What I don't ...
-1
votes
1answer
29 views

Alegbra Equation to solve client's request

A client of mine gave me an equation to solve withholding taxes and etc. This is what he gave me: $$A = \frac{B}{(P + B) / 1.12}$$ I do not see how to rearrange this into the form of $B=\ldots$. It ...
0
votes
2answers
33 views

An equation with a parameter

Given the equation $(|x+1|+|x-a|)^2-2(|x+1|+|x-a|)+4a-4a^2=0$ find all possible $a$ such that this equation has only one solution. I wanted to solve it like this: ...
1
vote
0answers
20 views

How Euclidian Algorithm for division works with algebric expressions?

I am attending an introductory Number Theory class for Computer Science focused on cryptography. I have done some exercises with integers number but I have two exercises in which appears algebric ...
4
votes
2answers
114 views

Solve this logarithmic equation.

$2\log_6(x^{1/2}+x^{1/4}) = \log_4 x$ I don't have any good ideas how to start. Im stuck at this easy question, I don't know what is going on.
4
votes
4answers
88 views

If $3^x +3^y +3^z=9^{13}$.Find value of $x+y+z$

Problem: If $3^x +3^y +3^z=9^{13}$.Find value of $x+y+z$. Solution: $3^x +3^y +3^z=9^{13}$ $3^x +3^y +3^z=3^{26}$ I am unable to continue from here. Any assistance is appreciated. Edited ...
0
votes
0answers
18 views

Finding the exact real roots of a system of a sinusoidal and a line

Say I have a sinusoidal function $s(x)=\alpha \sin(\beta x - \gamma) + \delta$ and the linear function $f(x)=mx+b$. How can I find $x$ exactly such that $s(x)=f(x)$? I can't solve it like a normal ...