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
6answers
63 views

How to find the roots of $-x^3+3x^2-7x+5 = 0$?

I would like to understand how to go about solving something like this, not just get the solution but some kind of methodology (that hopefully makes as much intuitive sense as possible); I honestly ...
3
votes
1answer
46 views

Nice polynomial reducibility: $x^n+4$

Problem: Find all $n\in \mathbb{N}$ such that $f(x)=x^n+4$ is reducible in $\mathbb{Z}[x]$. It seems $n=4k$ is the only one (the factorization follows easily from Sophie Germain's identity in this ...
0
votes
1answer
19 views

Exponential/Logarithmic Inequality

Being stated as the answer to a certain problem in a physics textbook the following inequality is implied: $$ \ln{T} ≤ \mu\theta + C \implies T ≤ T_{0}e^{\mu\theta} $$ It is also stated that as an ...
2
votes
4answers
81 views

Solving $e^{4x}+3e^{2x}-28=0$

How to solve this equation: $$e^{4x}+3e^{2x}-28=0$$ I don't know how to solve this problem. I read over another example, $e^{2x}-2e^x-8=0,$ and it said that $e^{2x}$ is $e$ to the $x$ squared, ...
0
votes
0answers
18 views

Sequence of integers in arithmetic progression and convergent sequence

Let $(x_n)_{n\geq1}$ be a sequence of integers. Define $y_n=\frac{x_n}{n},n\geq1$. The sequence $(y_n)_{n\geq1}$ is convergent and $n$ divides the sum of any $n$ consecutive terms of the sequence ...
0
votes
3answers
44 views

Solving a system of three equations: $d = s\cdot 3, c = s\cdot 1.5, c = 2\cdot d$.

Sorry if this is not the right place for this sort of question, but I am at a lost. My niece has some summer homework, and neither of us have a clue how to solve this question. Its been too long since ...
0
votes
2answers
57 views

How does $\sqrt {\frac{{4 + \sqrt {15} }}{8}} = \frac{{\sqrt {8 + 2\sqrt {15} } }}{4}$

I have the follow answering to a question from my textbook: $\sqrt {\frac{{4 + \sqrt {15} }}{8}}$ However my textbook simplifies it to: $\frac{{\sqrt {8 + 2\sqrt {15} } }}{4}$ I've checked and my ...
1
vote
2answers
19 views

Rewriting a trigonometric inequality (including a parameter)

How is it possible to rewrite these equations? $\sin{x}- \cos{x} ≤ \mu(\cos{x} + \sin{x}) \implies \tan{}x ≤ \frac{1 + \mu}{1 - \mu}$ and $\cos{x}- \sin{x} ≤ \mu(\cos{x} + \sin{x}) \implies \tan{}x ...
1
vote
4answers
78 views

How many solutions $k>1$ does the equation $\exp ((k-1)/( k+1))=\sqrt{k}$ have?

I have the following equation: $e^{\frac{k-1}{k+1}}=\sqrt{k}$. The question is: how many solutions does it have? ($e$ is Euler's constant and k is a positive real number >1).
1
vote
2answers
43 views

Having trouble solving $\log (x − 21) = 2 − \log x$ for $x$

I'm having trouble with this problem: $\log (x − 21) = 2 − \log x$, solve for $x$. I'm coming up with $x=-5$ but that can't be right.
-2
votes
1answer
37 views

Absolute values and Inequalities.

$$|3x - |x+1|| ≥ 2 $$ And $$|\frac{3}{5}x - 2| < 4 < \frac{x^2 + 6x - 4 }{ x + 1}$$ I hope that makes sense. Is my first time asking a Question. To questions. Need to find x in both of ...
0
votes
0answers
29 views

$\sum$ of binomial coefficients inequality

Let $m,n$ be positive integers with $m>n$. When is it true that $$m\cdot 5^{m-1}\cdot 3+\binom{m}{3}\cdot 5^{m-3}\cdot 3^3\cdot 2+\cdots +\binom{m}{2k+1}\cdot m^{m-2k-1}\cdot 3^{2k+1}\cdot ...
1
vote
2answers
57 views

Neither $\log x$ nor $\exp(x)$ are rational functions [on hold]

(a) Prove that $\log x$ cannot be expressed in the form $f(x)/g(x)$ where $f(x)$ and $g(x)$ are polynomials with real coefficients. (b) Prove that $e^x$ cannot be expressed in the form $f(x)/g(x)$ ...
1
vote
1answer
70 views

Find parameters of short geometric series [on hold]

I have 4 related numbers: Base (B) = 100 , Start (S) = 0.4 , Count (C) = 4 , Multiplier (M) = 0.64448 CASE 1: (B) 100 × (S) 0.4 = 40 (rr) (rr) 40 × (M) 0.64448 = 25.7788368 (r1) .... (counting 1) ...
0
votes
1answer
43 views

Solve $\frac{(x - 1)^3(x + 1)^8}{(x + 2)^4} > 0$

Solve the inequality $$\frac{(x - 1)^3(x + 1)^8}{(x + 2)^4} > 0$$ A) $X<1$ B) $X>1$ C) $X>-1$ D) $X<-1$ E) $X>-2$
1
vote
2answers
34 views

Finding (or rather expanding) the product $(5-xy)(3+xy)$

Given the product $(5-xy)(3+xy)$ I tried the following, As we know, $(x+a)(x+b)=x^2+(a+b)x+ab$ Here $x$ is $xy$. But $xy$ has two signs$-$ and $+$. How do I solve this.
0
votes
3answers
38 views

Finding the tangent line to the graph of $f(x)=(x+2)^{3/5}$ at $x=-2$

Does the graph of the function $f$ have tangent line at the given points? If yes, what is the tangent line? $f(x)=(x+2)^{3/5}$ at $x=-2$ solution: yes, $x=-2$ The derivative I found: ...
-1
votes
1answer
45 views

Finding two sided bounds on $(x+y)/(xy)$ given inequalities for $x$ and $y$

Given $\dfrac{1}{6} < x < \dfrac{1}{2}$ and $\dfrac{1}{7} < y < \dfrac{1}{3}$, can we determine bounds for $\dfrac{x+y}{xy}$?
-3
votes
1answer
22 views

Fourth roots of a certain complex number [on hold]

Find the fourth roots of $81(\cos 320^\circ + i\sin 320^\circ )$. Write the answer in trigonometric form. \begin{array} \text{a.} & 3(\cos 160^\circ + i \sin 160^\circ ); & &3(\cos ...
0
votes
2answers
18 views

Average Value - Graphs

long method: Determine an equation for each and solve using average value formula alternative methods? How could you prove the average value to be C over an interval [a,b] if you are given a ...
0
votes
1answer
8 views

Left & Right Area Approximation Using Y-Axis - Method Alternatives

Is there a simpler way of solving this then calculating x1(h)+x2(h)+x3(h)+x4(h) by using the given y values (in this case h, the height is one, because the length of each rectangle is one) ...
0
votes
1answer
9 views

Related Rates of Change - Cylinder Question

A cylindrical tank with radius 5 cm is being filled with water at rate of 3 cm^3 per min. how fast is the height of the water increasing? I dont want this question solved, but please help me correct ...
4
votes
4answers
62 views

How does $x^3 - \sin^3 x$ become $x^3 + \frac{1}{4}\sin{3x}-\frac{3}{4}\sin x$?

I was going through answers on this question and came across this answer and I was wondering how the user arrived at the first line where they state: $$f(x) \equiv x^3 - \sin^3 x = x^3 + {1 \over 4} ...
0
votes
3answers
56 views

Can anyone help me understand the simplification of $\frac{\sqrt 3 + \sqrt 2}{\sqrt 3 - \sqrt 2}\;$?

Can anyone help me understand the following simplification of the fraction? $$\dfrac{\sqrt 3 + \sqrt 2}{\sqrt 3 - \sqrt 2} = 5 + 2\sqrt 6$$ I cant understand how to simplify the left-hand side to get ...
2
votes
2answers
121 views

solving the inequality

I'm looking for hints on how to efficiently solve this inequality: $$\left( \frac {|x|-|1-x|}{|x|} \right)^{2x-1} \gt \left(\frac {|x|-|1-x|}{|x|} \right)^{8-x} $$
0
votes
1answer
18 views

Find equation of the straight line tangent to the curve at the point indicated

Find equation of the straight line tangent to the curve at the point indicated: $y=2x^2 -5$ at $(2,3)$ I think I have to use $y=m(x-x_o)+y_0$ etc but I'm not sure how to find the $m$! Thanks for ...
2
votes
3answers
37 views

Finding the perimeter of the room

If the length and breadth of a room are increased by $1$ $m$, the area is increased by $21$ $m^2$. If the length is increased by $1$ $m$ and breadth is decreased by $1$ $m$ the area is decreased by ...
0
votes
3answers
21 views

Finding the angles of a parallelogram.

In a parallelogram, one angle is $2/5th$ of the adjacent angles. Determine the angles of the parallelogram. I tried the following, Let the adjacent angles be $2x$ Let the other angle be $y$ ...
0
votes
2answers
24 views

The perimeter of a rectangle is 48 meters and its area is 135 m^2. Determine the sides of the rectangle.

The perimeter of a rectangle is 48 $m$ and its area is $135$ $m^2$. Determine the sides of the rectangle. I tried the following, Perimeter$=$$48$ $m$ Let the length be $x$ m and the breadth be $y$ m ...
3
votes
2answers
59 views

Mr. and Mrs. Ahuja weigh x and y kg. Find their present weights.

Mr. and Mrs. Ahuja weigh $x$ kg and $y$ kg respectively. They both take a dieting course at the end of which Mr. Ahuja loses $5$ kg and weighs as much as the wife weighed before the course. Mrs. Ahuja ...
1
vote
3answers
77 views

Solve the equation: $\frac{z}{z-5}+\frac{1}{3}=-\frac{5}{5-z}$

Solve the equation: $\frac{z}{z-5}+\frac{1}{3}=-\frac{5}{5-z}$ First $z$ cannot be equal to $5$. First, I multiplied $z$ with $3$, $1$ with $z-5$ and $-5$ with both. Eliminating the denominators ...
0
votes
0answers
58 views
+50

Proof of Descartes' theorem

I came across the use of Descartes' theorem while solving a question.I searched it but I could only find the theorem but not any ...
0
votes
1answer
35 views

The denominator of a fraction is 4 more than twice the numerator. Determine the fraction.

The denominator of a fraction is $4$ more than twice the numerator. When both the numerator and denominator are decreased by $6$, the denominator becomes $12$ times the numerator. Determine the ...
3
votes
0answers
47 views

How to find $f$ and $g$ if $f\circ g$ and $g\circ f$ are given?

The question is: Let $f:\mathbb R\rightarrow \mathbb R$ and $g:\mathbb R\rightarrow \mathbb R$ be two functions such that $(f\circ g)(x)=4x^2+4x+1$ and $(g\circ f)(X)=x^2+2x+2$. Find $f(x)$ and ...
0
votes
1answer
48 views

Find $\sin(x+y)$, given $\tan x$ and $\cos y$

Given that $\tan x= -2$ and $\cos y= 1/2$ where $x$ and $y$ are in the 4th and 1st quadrants respectively. Find, without evaluating angles $x$ and $y$, a) $\sin (x+y)$ Here is what i have done so ...
3
votes
2answers
54 views

Prove that every non-prime natural number $ > 1$ can be written in the form of $n+(n+2)+(n+4)+…+(n+2m) = p$

I'm trying to prove that every non-prime natural number greater than $1$ can is equal to a sum of consecutive even or odd numbers. This can be resumed in : « $p,m,n \in ℕ$» , «$p > 1$» , «$n > ...
2
votes
0answers
52 views

Eliminate variable in trigonometry equations

Say you have the equations: \begin{align} -S_1\sin\left(2\psi+\theta\right)+S_2\cos\left(\psi\right)&=S_3\\ S_1\cos\left(2\psi+\theta\right)+S_2\sin\left(\psi\right)&=S_4 \end{align} or ...
1
vote
3answers
53 views

Solve the following equation: $\frac{1}{x^2}+\frac{1}{(4-\sqrt{3}x)^2}=1$

Solve the following equation: $$\frac{1}{x^2}+\frac{1}{(4-\sqrt{3}x)^2}=1$$ I know it's from a Math Olympiad but I don't know which and I couldn't find it on the internet. Expanding everything ...
2
votes
3answers
28 views

Why $ (1- \sin \alpha + \cos \alpha)^2 = 2 (1 - \sin \alpha)(1+ \cos \alpha)$?

Why $ (1- \sin \alpha + \cos \alpha)^2 = 2 (1 - \sin \alpha)(1+ \cos \alpha)$? I am learning trigonometric identities one identity I have to proof is the next: $ (1- \sin \alpha + \cos \alpha)^2 = ...
0
votes
3answers
36 views

Derivation of sine and cosine case

I am struggling to see this. I know that we can factor out $ a$, but I don't see how we can end up with the right hand side. $$a \cos ^2(a t)-a \sin ^2(a t)=a \cos (2 a t)$$
0
votes
3answers
53 views

Find the values of parameter $x$ for which two graphs have 0, 1, or 2 intersection points

I have been given this question: $$y=\frac{4}{10}x+c\quad;\quad y=\frac 4x,$$ Investigate the values of $c$ that may provide, $0,1,$ or $2$ points of intersection. I'm really stuck, any ideas?
2
votes
4answers
56 views

Show that there is an angle $\theta$ such that $a=\cos\theta$ and $b=\sin\theta$

My problem is from Israel Gelfand's Trigonometry textbook. Page 50. Exercise 3: Suppose that $\alpha$ is some angle. If $a=4\cos^3\alpha-3\cos\alpha$ and $b=3\sin\alpha-4\sin^3\alpha$, show that ...
-2
votes
1answer
69 views

For all $x$ in $[0,90]$ show that $\cos(\sin x ) >\sin(\cos x )$

For all $x$ in $[0,90]$ show that $\cos(\sin x ) >\sin(\cos x )$ I understood the solution given in my book which said  $$\cos(x)+\sin(x)\leq\sqrt{2}<90$$ $$\cos(x)<90-\sin(x)$$ But if ...
1
vote
1answer
38 views

A farmer sold a calf and a cow for Rs 760. Find the cost of each.

A farmer sold a calf and a cow for Rs. 760 Thereby making a profit of 25% on the calf and 10% on the cow. By selling them for Rs. 767.5 he would have raised a profit of 10% on the calf and 25% on ...
1
vote
3answers
49 views

How many even 3 digit integers greater than 700

How many even 3 digit integers greater than 700 with distinct non zero digits are there ? My answer is: the only hundred digit that are possible are 7, 8 and 9 (3) the only ten digit that ...
0
votes
4answers
109 views

How to calculate $k^0+k^1+k^2 + k^3+…+ k^{n-1}$ [duplicate]

How to simplify below expression or convert it to something simpler like $k^{n-1}$? $$ k^0+k^1+k^2 + k^3+...+ k^{n-1} $$
1
vote
1answer
126 views

Showing that planes intersect

let there be two planes $$2x-y-5z+11=0$$ and$$2x+2y+z-1=0 $$ show that they intersect attempt at a solution: If planes do not intersect they are parralel hence there is a $t\in R$ such that ...
3
votes
1answer
20 views

Intersection of a circumference with a the curve: $y=ax^k$

Given the circunference centered in the origin of a cartesian reference frame, its equation is: $x^2+y^2=r^2$, Assuming $r=1$, we have: $x^2+y^2=1$. The intersections of this curve with the curve ...
0
votes
1answer
17 views

Knowing the total price of stamps of two denomination, find the number of stamps of each kind

A man buys postage stamps of denominations $3$ paise and $5$ paise, for Rs $1$. He buys $22$ stamps in all. Find the number of $3$ paise stamps bought by him. (100p= 1 Rs) I tried, Let the number of ...
3
votes
1answer
26 views

Sketch The Region In The Plane Defined By $\lfloor x + y\rfloor^2 = 1$

Sketch The Region In The Plane Defined By $\lfloor x + y\rfloor^2 = 1$ I would like for you guys to have a look at my approach and give my advice regarding the solution and whether there's a ...