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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2
votes
2answers
231 views

Integration: Method of partial fractions - Any standard method of finding constants in hard to solve expressions?

I've been computing many indefinite integrals using the method of partial decomposition. The integrals are usual on the form $$\int \frac {x^2-29x+5} {(x-4)^2(x^2+3)} dx$$ which is equal to $$\int ...
2
votes
9answers
110 views

Why is $\log\frac{1}{2} = -\log(2)$

Why does $\log\frac{1}{2} = -\log(2)$ What rule is being used? EDIT: Wow, that was fast. Thanks for the replies. I saw it shortly after I posted it.
5
votes
2answers
231 views

Solving $x^2 - 1 = e^x$

Can someone help me solve the equation $x^2 - 1 = e^x$ ? I tried taking the natural logarithm of both sides but I don't know where to go from there.. I got: $\ln(x^2 -1) = x$ But I don't know how ...
0
votes
3answers
34 views

Basic Arithmetic Questions

How does one solve the following: and also $(x^2 - 4x +4)^x + (2-x)^x <2 $ ? Should logarithms be used here or there should be some algebraic method which takes into account the proprieties of ...
7
votes
1answer
121 views

How do we solve this system of equations?

$a,b \in \Bbb R$ and $$\frac{a^5b-b^5a}{a-b}=30$$ and $$a^5+b^5 = 33$$ I get that $a^6-b^6=(a-b)63$ But I have no idea how to solve after that. Someone could help me?
1
vote
1answer
49 views

Find out the design of a cylinder

A cylindrical can is made from tin.If it can be contain $1000 m^3$ liquid inside it then what is the parameter of design if we are oblige use the minimum amount of tin. My teacher give me this and say ...
2
votes
1answer
78 views

Showing $(a+b+c)(x+y+z)=ax+by+cz$ given other facts

$$x^2-yz/a=y^2-zx/b=z^2-xy/c$$ None of these fractions are equal to 0.We need to show that, $(a+b+c)(x+y+z)=ax+by+cz$ This question comes from a chapter that wholly deals with factoring ...
0
votes
1answer
40 views

Prove that there exist two infinite sequences that simultaneously satisfies all these conditions

Prove that there exist two infinite sequences $\langle a_n\rangle_{n\geq 1}$ and $\langle b_n\rangle_{n\geq 1}$ of positive integers such that the following conditions hold simultaneously: $$1 < ...
0
votes
1answer
55 views

simple arithmetic question

Trying to solve the following inequality numerous times I've been reaching a wrong solution time after time: $$(x^2 - 4x +4)^x + (2-x)^x <2 $$ after setting all the demands using systems of ...
1
vote
0answers
12 views

Fourier analysis of real valued function

Under what condition is it not possible to obtain the fourier transform of a real valued function?
1
vote
2answers
172 views

Simplify $(6\sqrt{x} + 3\sqrt{y})\cdot(6\sqrt{x} - 3\sqrt{y})$. [closed]

This expression $(6\sqrt{x} + 3\sqrt{y})\cdot(6\sqrt{x} - 3\sqrt{y})$ equals: ? Simplify the expression. I'm tried multiplying but I'm not getting the right answer.
1
vote
6answers
252 views

Find $(a,b)$ such that in $x^2+ax+b$, whenever $v$ is a root, then $v^2 - 2$ is also a root

Find $(a,b)$ such that in $x^2+ax+b$, whenever $v$ is a root, then $v^2 - 2$ is also a root $a,b$ are real numbers. Roots may or may not be real. In this question, the aim is to find values of and b ...
0
votes
1answer
29 views

Prove, for these set of constraints, $\sum_{i=1}^k \frac{x_i}{i} \leq 3$

Suppose $\left\langle x_1, x_2, \dots, x_n, \dots\right\rangle$ is a sequence of positive real numbers such that $x_1 \geq x_2 \geq \dots \geq x_n\dots$ and for all $n$: $$\sum_{i = 1}^n ...
1
vote
4answers
231 views

Proof: Two polynomials $P(x)$ and $Q(x)$ attain same value for every $x \in \mathbb R$ if and only if coefficients $p_i = q_i$ are equal for every $i$

Proof: Two polynomials $P(x)$ and $Q(x)$ attain same value for every $x \in \mathbb R$ if and only if coefficients $p_i = q_i$ are equal for every $i$ I've been thinking how to prove this. I know we ...
1
vote
1answer
67 views

How do you solve this trig equation, I've tried what I know..

How do I solve this question? $$2\sin{x} + \cos{x} = 0$$ I tried taking $\cos{x}$ from each side, and dividing through to make $-2\tan{ x }= 0$, but then I got stuck. I probably worked it out ...
1
vote
2answers
78 views

Prove $a = b = c$, given $P_1(x) = ax^2-bx-c$ , $P_2(x) = bx^2-cx-a$, $P_3(x)=cx^2-ax-b$ and $P_1(v)=P_2(v)=P_3(v)$

Prove $a = b = c$, given $P_1(x) = ax^2-bx-c$, $P_2(x) = bx^2-cx-a$, $P_3(x)=cx^2-ax-b$ and $P_1(v)=P_2(v)=P_3(v)$ where $v$ is a real number. $a,b,c$ are non zero real numbers.
0
votes
1answer
53 views

Adjust Saturation in CIE L*a*b* space.

Given a color in CIE L*a*b* space, how does one change the saturation? This is what I know... $$\mathrm{chroma} = \sqrt {(a^*)^2 + (b^*)^2}$$ $$\mathrm{hue} = \arctan \left( a^* \over b^* \right)$$ ...
0
votes
3answers
48 views

Unequal distribution of n things among p persons

This is from Higher Algebra by Hall & Knight. In how many ways can $n$ things be given to $p$ persons, when there is no restriction as to the number of things each may receive? Answer: ...
0
votes
2answers
140 views

Show that $a<b$ iff $a^n<b^n$

At first I thought it was obvious, but one implication is giving me a hard time. I would appreciate if one the implications demonstrated were to be revised as well help or hints with the other ...
-1
votes
3answers
47 views

Can someone help me solve this problem dealing with area of a sector/circles?

If a pizza is $1/3$ of an inch thick and has a diameter of $8$ inches, how many cubic inches of pizza have you eaten if you eat a sector whose edges form a $20^{\circ}$angle? Answer to the nearest ...
0
votes
4answers
141 views

How do you solve the diference quotient for $f(x) = 23 \sqrt{x}$

How do I solve the difference quotient for $f(x) = 23 \sqrt{x}$? I know how to plug it in but I don't understand how to simplify.
1
vote
1answer
77 views

Trignometric Identities

Question 1: Develop a formula for $\sin (x/2)$ in terms of $x$. Question 2: Use a double angle formulae to develop a formula for $\sin 4x $ in terms of $x$. I have absolutely no idea how to do ...
0
votes
1answer
105 views

Complex Numbers Question?

I answered the first part of the question. But I'm having a trouble with the second part. I can only find the half-line at $2i$ and $\theta=\pi/6$. Here's the solution guide:
1
vote
0answers
239 views

Question on Proof of Shoelace Formula

I was looking for a way to prove the shoelace formula when I found this proof: For this clockwise order to make sense, you need a point O inside the polygon so that the angles form $OA_{i}A_{i+1}$ ...
2
votes
4answers
119 views

Finding the GCD of $50!$ and $2^{50}$

I've been trying to figure out how $n!$ and $x^n$ are related (where x is an integer) for most of the morning - I know it must be the key to unlocking this problem. Up to this point I've only used ...
1
vote
1answer
75 views

Graphing the function $f(x) = \frac{(x-1)(x-2)}{(x(x+2)(x-3))}$

When I (try to) graph the function $f(x) = \frac{(x-1)(x-2)}{(x(x+2)(x-3))}$ I start by finding the vertical asymptotes $x = 0$, $x = -2$ and $x = 3$ and the roots $x = 1$ and $x = 2$. At this point ...
1
vote
2answers
308 views

Inequality with cube roots

$$(\sqrt{n}+1)^{1/3}-(\sqrt{n}-2)^{1/3} \geq (\sqrt{n+1}+1)^{1/3}-(\sqrt{n+1}-2)^{1/3}$$ $$n\in \mathbb{N}$$ I come upon this inequality when trying to use prove series declension for the Leibnit'z ...
0
votes
3answers
39 views

How to find the expresion such that its derivative must meet a certain condition

Suppose $a$ and $b$ are expressions in terms of the variable $x$. We know: $\begin{align} a &= a \cdot\frac{b}{b} \\ &= \frac{ab}{b} \\ \end{align} $ Is there a systematic way to find $b$ ...
1
vote
2answers
126 views

Evaluation Integral: Method of Partial Fractions - Can the system of equations always be solved?

I've been studying the method of partial fractions for evaluation integrals. So far every example and exercise have been fairly straight forward, but I still have some unanswered questions: 1) The ...
3
votes
2answers
64 views

Show that $n-kl$ is a perfect square

I faced a doubt in this question while solving some maths problem. Please, solve it. A natural number $n$ is chosen strictly between two consecutive perfect squares. The smaller of these two square ...
1
vote
2answers
68 views

exponential equation with a sum of exponents

I'm trying to solve the following exponential equation: $e^{2x} - e^{x+3} - e^{x + 1} + e^4 = 0$ According to the the text I am using the answer should be $x = 1,3$ but I can't derive the ...
1
vote
2answers
88 views

Why can't you solve this probability problem in this way?

Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three ...
2
votes
5answers
86 views

How to notice that $3^2 + (6t)^2 + (6t^2)^2$ is a binomial expansion.

The other day during a seminar, in a calculation, a fellow student encountered this expression: $$\sqrt{3^2 + (6t)^2 + (6t^2)^2}$$ He, without much thinking, immediately wrote down: $$(6t^2+3)$$ What ...
0
votes
1answer
70 views

$x^3=a+1,x+{b\over x}=a,x=?$

$x^3=a+1,x+{b\over x}=a,x=?$ I did in usual manner of finding root of quadratic equation, but got roots with surd factors, but here are the options a) $a(b+1)/(a^2-b)$ b) $(ab+1)/(a^2-b)$ c) ...
2
votes
3answers
175 views

$(ab)^2=(bc)^4=(ca)^x=abc$ Then what is the value of $x$?

Given that $(ab)^2=(bc)^4=(ca)^x=abc$ Then what is the value of $x$? $2(\log a+\log b)=4(\log b+\log c)=x(\log c+\log a)=\log a+\log b+\log c$ Then I am lost, any other easier way to solve?
6
votes
6answers
639 views

Why can't I simply use algebra to solve this inequality?

Consider the inequality: $\frac{(x+3)(x-5)}{x(x+2)}\geq 0$ Why can't I simply multiply both sides by $x(x+2)$ and get $(x+3)(x-5)\geq 0$ ? Which would yield: $x^2-2x-15\geq 0$ and I could then use ...
-2
votes
4answers
158 views

mental ability whiz

I got a difficult question in a national olympiad, and was not able to solve it. I can't wait for answer keys. please solve it for me! If $3a = 4b = 6c$ and $a + b + c = 27 \sqrt{29}$, then what is ...
1
vote
2answers
54 views

$a,b\in\mathbb{R}\ni a<b<{1\over a}$ and $x=(a+{1\over a})-(b+{1\over b})$

$a,b\in\mathbb{R}\ni a,b>0, a<b<{1\over a}$ and $$x=(a+{1\over a})-(b+{1\over b})$$ Then $a) x>0$ b) $x<0$ $c) x=0$ d) no such conclusion can be drawn about $x$ just confirm me ...
2
votes
3answers
123 views

How many ordered triples of integers which are between 0 and 10 inclusive do we actually have if $a * (b+c) = a * b +c$

How many ordered triples of integers $(a,b,c)$ which are between 0 and 10 inclusive do we have if: $a * (b+c) = a * b +c$
0
votes
2answers
285 views

Algebraic Manipulation question - trying to get alternate form

I'm currently working on algebraic manipulation, changing algebraic fractions into a chosen alternate form but I've hit a brick wall. I'm trying to get: $$\frac{2(3^x - 2^x)}{3^{x+1} - 2^{x+1}}$$ ...
0
votes
1answer
29 views

Solving for $x$ in an equation involving rational powers

$128000-256x^{3/4}\left(\frac{256}{625x}\right)^{1/4}=0$ I need help doing the problem. The answer is $625$. I started off by isolating $128000,$ but now I'm stuck.
0
votes
2answers
55 views

Simplify equation to isolate variable by itself

I feel really dumb for not being able to figure this out: 12/x = (.1 + (.08/10) + 1 + .08) How can I get this equation to have ...
0
votes
1answer
49 views

Solving equation in terms of $x$

$x + .08x + (x +.08x).1 = 12$ I need to come up with a formula that solves for x. In a generic way that doesn't combine values of $x$. For example: starting with $x + .08x = 12$ $$x = \frac{12}{1+ ...
2
votes
2answers
136 views

Can this algebraic equation of degree 5 be solved?

I have the following algebraic equation of degree 5 which I would like to solve for $x \in \mathbb{R}$: $$f(x) =ax^3 +bx^2 + cx + d \text{ with } a \in \mathbb{R}_{>0},\; b,c,d,w,z \in ...
1
vote
3answers
125 views

De Moivre's formula to solve $\cos$ equation

Use De Moivre’s formula to show that $$ \cos\left(x\right) + \cos\left(3x\right) = 2\cos\left(x\right)\cos\left(2x\right) $$ $$ \mbox{Show also that}\quad \cos^{5}\left(x\right) = ...
0
votes
1answer
52 views

Given $\frac{y+a}{x+a}=b$, is there a solution for $\frac{x}{y}$?

I have an expression of the form: $$\frac{y+a}{x+a}=b$$ I know there is an infinity of solutions for x and y, but what I'm looking for is a solution for x/y, unique or otherwise.
3
votes
2answers
70 views

Can $a^2+b+2$ and $b^2+4a$ both be perfect squares?

Are there any positive integers $a$ and $b$ so that $a^2+b+2$ and $b^2+4a$ are both perfect squares?
0
votes
1answer
65 views

Please help me with factorisation

Is it possible to write $$64x^6-112x^4+56x^2-7$$ in linear factors? If so, what are they? (Finding it really difficult to ask this question!!)
2
votes
1answer
209 views

Factorizing $(x-1)(x-3)(x-5)(x-7)-64$

We need to factorize: $$(x-1)(x-3)(x-5)(x-7)-64$$ We can, by the rational root theorem, see that there are no roots of this polynomial.Next observation is that $64=(8)^2$. So this means that if the ...
3
votes
3answers
145 views

Proving $\csc (x) +\cot( x)=\frac{\sin (x)}{1-\cos(x)}$

I have this problem: Prove that $\csc (x) +\cot( x)=\dfrac{\sin (x)}{1-\cos(x)}$ From LHS I tried using $\sin^2x+\cos^2x = 1$ and ended up nowhere. I tried rearranging RHS but ended up with ...