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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3
votes
4answers
371 views

Solve $2^{x}=x^{2}$

I've been asked to solve this and I've tried a few things but I have trouble eliminating x. I first tried taking the natural log: $x\ln \left( 2\right) =2\ln \left( x\right) $ $\dfrac {\ln \left( ...
0
votes
2answers
73 views

Basic math question

I'm going back to school and haven't taken a math class in years, so I'm brushing up on the basics. The text states $\frac{g(t + \Delta(t))^2}{2} = \frac{gt^2}{2} + \frac{g}{2}\left(2t\Delta t + ...
3
votes
2answers
257 views

Proving $p\nmid \dbinom{p^rm}{p^r}$ where $p\nmid m$

A question from Advanced Modern Algebra by Joseph J.Rotman. Let $n=(p^r)m $ such that the prime $p\nmid m$.Prove that $p\nmid \dbinom{n}{p^r}$.HINT: Assume otherwise,cross multiply and apply ...
1
vote
0answers
28 views

Finding the gradient from two points

I've come across the following question in a text book: Find the gradient of the line joining the following pair of points: (p+3, q-7), (p+5, 3-q) So I did the following: y2-y1 -> 3-q-q+7 ...
2
votes
8answers
524 views

Simplify $2^{(n-1)} + 2^{(n-2)} + … + 2 + 1$

Simplify $2^{(n-1)} + 2^{(n-2)} + .... + 2 + 1$ I know the answer is $2^n - 1$, but how to simplify it?
0
votes
2answers
102 views

What is my overall grade percentage score?

I received 20/30 for one assignment worth 30% of the overall mark. 21/30 for the second, which is worth 30% of the overall mark. And 15.5/40 for the third which is worth 40%. What is my overall grade ...
2
votes
2answers
46 views

Finding $x$. The summation of the floor of the equation.

I would appreciate if somebody could help me with the following problem Q:Finding $x$. The summation of the floor of the equation. $$\sum_{i=1}^{2013}\left\lfloor\frac{x}{i!}\right\rfloor=1001$$
0
votes
2answers
36 views

Percentages and Proportions

Given two integer variables $x$ and $y$. We are given that each integer variable $x$ and $y$ can't be greater than a given integer $z$. The problem: We are given the proportions $a$ and $b$ such that ...
1
vote
4answers
61 views

Integer $a$ , If $x ^ 2 + (a-6) x + a = 0 (a ≠ 0)$ has two integer roots

If the equation $x ^ 2 + (a-6) x + a = 0 (a ≠ 0)$ has two integer roots. Then the integer value of $a$ is $\bf{My\; Try}::$ Let $\alpha,\beta\in \mathbb{Z}$ be the roots of the equation . Then ...
0
votes
4answers
612 views

2 Easy GRE questions

I've been having trouble with these two questions. The first is simple interest, the second is rate. I'm sure they're easy but I can't focus on getting the solution because I'm terrible at focusing on ...
7
votes
3answers
112 views

Solve the equation $\lfloor x^2\rfloor-3\lfloor x \rfloor +2=0$

Solve the equation $$ \lfloor x^2\rfloor-3\lfloor x \rfloor +2=0 $$ where $\lfloor x\rfloor $ denotes floor function. My Attempt: Let $x = n+f$, where $n= \lfloor x \rfloor \in ...
0
votes
1answer
102 views

Solve the inequality $2^{\left( x^{3}-x\right) } < 1$

$2^{\left( x^{3}-x\right) } < 1$ Let $2^{\left( x^{3}-x\right) }-1=f\left( x\right)$ To find the values for which $f(x)<0$ I let $f(x)=0$: $2^{\left( x^{3}-x\right) }-1=0$ $2^{\left( ...
2
votes
1answer
123 views

Precalculus - Exponential and Logarithmic Equations

Mike Kallenberg deposited some money in a bank account that earns 5.6% interest compounded continuously. How long would it take to double the amount in money in Mr. Kallenberg's account?
5
votes
2answers
89 views

Proof that b is not divisible by 6

$$b=\left \lfloor (\sqrt[3]{28}-3)^{-n} \right \rfloor$$ The brackets mean that the number is the largest integer smaller than $(\sqrt[3]{28}-3)^{-n} $ Proof that b is never divisible by 6. I have ...
0
votes
3answers
855 views

Irrational Roots To Third-Degree Polynomial Equations

How do I find irrational roots to a third-degree polynomial equation? I have already used up all my p/q options so there are no rational roots. I don't think I can use the quadratic formula because it ...
4
votes
1answer
66 views

Writing $\frac{(\sqrt{2}+1)^{2n+1}+(\sqrt{2}-1)^{2n+1}}{2\sqrt{2}}, n\geq2$ as sum of two perfect squares

I tried to show that $$ {\left(\sqrt{2\,} + 1\right)^{2n+1} + \left(\sqrt{2\,} - 1\right)^{2n+1} \over 2\,\sqrt{2\,}}\,,\qquad n\geq2 $$ is written as the sum of two perfect squares. We used Newton's ...
1
vote
1answer
29 views

3 equations with 3 unknowns

The Little Town Arts Center charges $\$23$ for adults, $\$12$ for senior citizens, and $\$8$ for children under 12 for their live performances on Sunday afternoon. This past Sunday, the paid revenue ...
0
votes
2answers
87 views

logarithms equations, different bases

solve equations: $\log_x 10 +2\log_{10x} 10-3\log_{100x} 10=0$ so I tried to use $\log_a b=\frac{1}{\log_b a}$ but it didn't work for me.
0
votes
4answers
112 views

How to prove that $(3+2\sqrt{2})^n=a_n+b_n\sqrt{2}$ for some positive integers $a_n,b_n$ without induction?

I have to prove that without induction: Let $n$ is non-negative integer number, prove that: $(3+2\sqrt{2})^n=a_n+b_n\sqrt{2}$ where $a_n, b_n$ are positive integer number My try: $a_1=3, b_1=2$ ...
2
votes
2answers
86 views

Equations with unknowns and powers [closed]

$a + b +c = 17$ $a^2 + b^2 + c^2 = 101$ $a^3 + b^3 + c^3 = 623$ How does one go about solving this? Thanks
49
votes
7answers
3k views

$\sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7\cdots}}}}}$ approximation

Is there any trick to evaluate this or this is an approximation, I mean I am not allowed to use calculator. $$\sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7\cdots}}}}}$$
3
votes
2answers
82 views

Proving by induction that $\sum_{i=2}^n(i^2-i) = \frac{n(n^2-1)}{3}$ for all $n \ge 2$

Doing proof by induction exercises, everything was fine until I tried to do one with $\sum$ : Prove that $$\sum_{i=2}^n(i^2-i) = \frac{n(n^2-1)}{3}$$ holds for all $n \ge 2$. Now, my ...
0
votes
1answer
725 views

Use fundamental identities to write the first expression in terms of the second, for any acute angle θ. sec(θ); sin(θ)

Can someone please help me with the question above? I don't understand what it's asking. A thorough explanation would be much appreciated.
1
vote
0answers
48 views

A problem on logarithm

Can this expression (see below) be written in the form $g_kw^{h_k}$. Where $g_k$ and $h_k$ are functions of only $k$?: $(1- k)^{\lceil\log_kw\rceil - 1}$. Here $k$ and $w$ are positive integers. I ...
19
votes
6answers
4k views

Can $x^3+3x^2+1=0$ be solved using high school methods?

I encountered the following problem in a high-school math text, which I wasn't able to solve using factorization/factor theorem: Solve $x^3+3x^2+1=0$ Am I missing something here, or is indeed a more ...
4
votes
1answer
56 views

Is there a commonly accepted notation for algebraic numbers?

In this question I needed a way to denote an algebraic number using a polynomial equation it satisfies and its isolating polynomial. Because I am not aware of any commonly accepted notation for this, ...
0
votes
4answers
120 views

What is easiest way to prove that $\sqrt 8/2$ is equal to $\sqrt 2$. [duplicate]

What is the most easy way I can prove $\sqrt 8/2$ is equal to $\sqrt 2$ I have done $(\sqrt 8/2)^2$ but at the end it gives me $\pm \sqrt 2$ and not positive $\sqrt 2$ So how?
5
votes
2answers
449 views

$\text{Let }y=\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5-…}}}} $, what is the nearest value of $y^2 - y$?

I found this question somewhere and have been unable to solve it. It is a modification of a very common algebra question. $\text{Let }y=\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5-...}}}} $, what is the ...
0
votes
2answers
37 views

Is this possible?

Is it possible to convert this forumla $$r(\theta) = \sum_{n=0}^\theta\left(\frac{2n+1}{2}\right)$$ to one without the $\sum$ sign? If so, how?
4
votes
1answer
140 views

Find zeros of function $f(x)$

If I have $$x^2(x-3)(x+3)=0$$ then the solutions are: $$x_{1,2}=0, x_3=3, x_4=-3 $$ or $$x_{1}=0, x_2=3, x_3=-3?$$ So are there 4 or 3 solutions?
0
votes
2answers
180 views

How to calculate profit share for this example [closed]

How can we calculate a profit share for Rs. 100,000 that remained in an account for 5 days only. See this question illustration below I was confused about which tags were more appropriate for ...
0
votes
3answers
46 views

Why do I receive the wrong answer when I try to solve this exponential equation?

So I have the equation: $25^{x}=5^{x}+6$ My reasoning is if you make everything to the base 5: $\left( 5^{2}\right) ^{x}=5^{x}+5^{\log _{5}6}$ Given the bases are the same we can do: $2x=x+\log ...
0
votes
2answers
693 views

Calculating monthly instalment after down payment

An item is available for $34000\$$ cash or $20000\$$ cash down payment together with $5$ equal monthly instalments. If the rate of interest charged under the instalment plan is $30\%$ per annum, ...
2
votes
1answer
75 views

Number of selection containing at least one of each kind.

From 3 cocoa nuts, 4 apples, and 2 oranges, how many selections of fruit can be made, taking at least one of each kind ? Ans:315 My thought: For any of our selection that contains at least one of ...
1
vote
3answers
102 views

Can someone help me simplify this trig expression?

$$( \tan x+ \sec x )( \cot x-\cos x ) $$ I got stuck after a few steps of converting and adding and what not.
2
votes
1answer
78 views

Find the minimum value of the expression.

Find the minimum value of the expression. $x,y,z \in R$ $\sqrt{x^2+1}+ \sqrt {4+(y-z)^2} + \sqrt{1+ (z-x)^2} + \sqrt{9+(10-y)^2}$
0
votes
3answers
64 views

Can someone please explain how this was factored?

How was $x^2(x+1)-4(x+1)$ factored into $$(x^2-4)(x+1)?$$ I know this seems very basic but can someone please explain this?
1
vote
1answer
76 views

Product of gradients of x=0 and y=0

A friend asked me this question: The product of the gradient of any two lines perpendicular to each other is $-1$. Now, the lines $x=0$ and $y=0$ are perpendicular to each other. If you take the ...
2
votes
3answers
70 views

Solve: $2\log_{3}(x)-\log_{3}(x+6)=1$

just getting going with logarithms, having trouble with this question. $$2\log_{3}(x)-\log_{3}(x+6)=1$$ $$\log_{3}x^2-\log_{3}(x+6)=1$$ Stuck at this point: What do I do next? ...
1
vote
3answers
59 views

Why is the quadratic equation $ax^2+bx+c=0$?

Shouldn't it be $y=ax^2+bx+c$? According to Wikipedia, it is $ax^2+bx+c=0$. I guess that they are both equations, right?
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?