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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4
votes
1answer
210 views

$a,b,c>0,a+b+c=21$ prove that $a+\sqrt{ab} +\sqrt[3]{abc} \leq 28$

$a,b,c>0,a+b+c=21$ prove that $a+\sqrt{ab} +\sqrt[3]{abc} \leq 28$ I have tried to use AM-GM inequality, but get no result as follows: $$a+\sqrt{ab}+\sqrt[3]{abc}\leq ...
0
votes
2answers
209 views

Converting squared or cubed units

At the moment I am going from square meters to square yards. One meter is $1.0936$ yards. So I figure $1\text{m}^2 = 1.0936 \text{yards}^2$ I know it isn't but I want to learn a good systematic ...
9
votes
2answers
1k views

If $x^2=y^2$, prove that $x=y$ or $x=-y$

I have a simple question here. I am trying to prove that, given $x^2=y^2$, $x=y$ or $x=-y$. I know exactly why this is true; it's obvious. I'm just unclear on the general format of a proof, as well as ...
12
votes
3answers
1k views

A real solution to a cubic equation

What is the easiest way to find the real solution of the equation $x^3-6x^2+6x-2=0$? I know the solution to be $x=2+2^{2/3}+2^{1/3}$ (Mathematica) but I would like to find it analytically. If ...
2
votes
2answers
135 views

Divide polynomials with exponents and simplify

The expression is $$\frac{p^2q^2}{m^2-n^2} \cdot \frac{m^2+2mn+n^2}{3p^2+2pq-q^2}.$$ How could I divide and simplify this?
0
votes
2answers
69 views

Algebra word problem

The population of a city in 1910 was 50,000, and it doubles every 10 years. What will it be (a) in 1970 (b) in 1990 (c) in 2,000? Im trying to brush up my algebra skills and found this question. I ...
1
vote
2answers
258 views

Multiplying and simplifying expressions

The expression is: $$\frac{24a^4b^2c^3}{25xy^2z^5} \cdot \frac{15x^3y^3z^3}{16a^2b^2c^2}$$ What I did was subtract the exponents of the numerator to the exponents of the denominator. I did a cross ...
0
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6answers
1k views

How to solve equations with square roots?

I'm struggling with an obviously easy problem: Find $x,y$: $I: \; \sqrt x + \sqrt y=8, \quad \quad II: \; \sqrt{xy}=15$ I tried different ways (put $\sqrt x$ from $I$ into $II$) to solve these ...
1
vote
2answers
3k views

Perform the indicated operations and simplify

$${2x^2+5x-3\over16x^2+26x+9}\cdot{10x^2+13x+4\over5x^2+11x-12}\cdot{35x^2-38x+8\over x^2+x-2}$$ Please help me. I'm having a hard time with 3 terms.
1
vote
1answer
73 views

Let $f(x)=a_0^2x^n+a_1x^{n-1}+a_2x^{n-2}+…+a_n,\,$ where $a_0,a_1,…,a_n \in \Bbb R$

I was stuck on the following problem which I came across during my study of theory of equations: Let $f(x)=a_0^2x^n+a_1x^{n-1}+a_2x^{n-2}+...........+a_n,\,$ where the coefficients ...
1
vote
1answer
53 views

Performing and simplifying equations

The expression is $$\frac {5}{x-2} - \frac 3{x+7} + 2$$ I need to simplify it too.
1
vote
2answers
139 views

Can we refer to the standard form of a quadratic equation as the general form as well?

I would like to know if we can refer to $$ax^2+bx+c=0$$ as the "general form" of a quadratic equation, or is it only called the standard form?
1
vote
1answer
64 views

Performing operations and simplification

The equation is $$2a-7b- \dfrac{4(a^2-16b^2)}{2a-3b}$$ I need to simplify the expression.
0
votes
1answer
247 views

Second differences in polynomials of degree 2 (Gelfand/Shen, Algebra. Chapter 38)

Problem 156 (page 72): $$ P(x)=x^2-x-4 $$ Prove that the second differences are not a coincidence and that they repeat for all $ x\geq 0 $. Problem 157 (page 73): Prove that for any ...
1
vote
2answers
77 views

Proving Inequality $\forall x\ge1$

How do I prove that $\forall x\ge1$ $$\left(\frac{x+1}x\right)^x\left(\ln\left\{1+\frac1x\right\}-\frac1{x+1}\right)>0\,?$$ I have tried rearranging this many times but I always end up with a ...
0
votes
1answer
33 views

Simplify the following problem

How $$\frac{1}{k}\sum_{m=r}^{k-1}\frac{m!}{(m-r)!}=\frac{(k-1)(k-2)\ldots(k-r)}{r+1};\quad r=1,2,\ldots$$ I have thought in two ways: ...
0
votes
1answer
21 views

Simple Calculation

How $$\sum_{m=r}^{k-1}m(m-1)\ldots(m-r+1)=\sum_{m=r}^{k-1}\frac{m!}{(m-r)!};\quad r=1,2,\ldots$$ ? i have ...
3
votes
3answers
1k views

Improving basic maths & algebraic manipulation skills

I am currently doing a-levels in the UK (received an A in AS) so doing calculus (differential equations/trig integrals)/linear algebra/mechanics/statistics etc. However I feel that a lot of my basic ...
21
votes
1answer
586 views

Is it always possible to factorize $(a+b)^p - a^p - b^p$ this way?

I'm looking at the solution of an IMO problem and in the solution the author has written the factorization $(a+b)^7 - a^7 - b^7=7ab(a+b)(a^2+ab+b^2)^2$ to solve the problem. It seems like it's always ...
0
votes
1answer
89 views

solving equations by the method of elimination

$$\frac{a}{x}+\frac{b}{y}=\frac{a}{2}+\frac{b}{3} \;\;\; \ldots(i)$$ $$x+1=y \;\; \;\ldots(ii)$$ We have to solve for $x$ and $y$, only this time using the method of elimination. From equation ...
1
vote
1answer
77 views

Summation of series

Write down the sum of $\displaystyle \sum_1^{2N} n^3$ in terms of $N$, and hence find: $1^3 - 2^3 + 3^3 - 4^3 + \cdots - (2N)^3$ in terms of $N$, simplifying your answer. I found this to be ...
0
votes
2answers
86 views

How To Evaluate This Limit and Find c?

For a certain value of $c$, $$ \lim_{x \to \infty}\left[\left(x^{5} + 7x^{4} + 2\right)^c - x\right] $$ is finite and non zero. Let $l$ be this limit. Calculate $l-2c$. The answer is $1$ but I am ...
1
vote
1answer
83 views

Complex polynomial identity with norm condition

In this question, the following was shown: If $R(z)=\dfrac{P(z)}{Q(z)}$, where $P,Q$ are polynomials in a complex variable $z$, satisfies the condition that $|R(z)|=1$ whenever $|z|=1$, then the ...
11
votes
4answers
1k views

How to show that $A^3+B^3+C^3 - 3ABC = (A+B+C)(A+B\omega+C\omega^2)(A+B\omega^2+C\omega)$ indirectly?

I found this amazingly beautiful identity here. How to prove that $A^3+B^3+C^3 - 3ABC = (A+B+C)(A+B\omega+C\omega^2)(A+B\omega^2+C\omega)$ without directly multiplying the factors? (I've already ...
1
vote
1answer
37 views

Use of polynomial with reciprocals

Let $P(z), Q(z)$ be polynomials, and define $R(z)=\dfrac{P(z)}{Q(z)}$, where $P(z)$ and $Q(z)$ have no common factors. Greater unity is achieved if we let the variable $z$ as well as the values ...
6
votes
3answers
180 views

If this is a telescoping series then how does it collapse? $\frac{3r+1}{r(r-1)(r+1)}$

Express $$\frac{3r+1}{r(r-1)(r+1)}$$ in partial fractions. Hence, or otherwise, show $$\sum_{r=2}^n\frac{3r+1}{r(r-1)(r+1)}=\frac52-\frac2n-\frac{1}{n+1}$$ So, I have obtained the partial fractions ...
0
votes
1answer
309 views

Explanation of the binomial theorem and the associated Big O notation

I'm currently following the MIT Single Variable lectures online and the professor states that the binomial theorem for the expansion $(x + \Delta x)^{n} = x^{n} + nx^{n-1}\Delta x + O((\Delta ...
0
votes
2answers
53 views

Using $\frac n6(n+1)(2n+7)$ find, interms of $n$, the sum of the series $3\ln2+4\ln2^2+5\ln2^3+…+(n+2)\ln2^n$

Using the result $$\sum_{r=1}^nr(r+2)=\frac n6(n+1)(2n+7)$$ find, interms of $n$, the sum of the series $$3\ln2+4\ln2^2+5\ln2^3+...+(n+2)\ln2^n$$ and express in its simplest form. Where do I start ...
4
votes
2answers
228 views

Trig identity problem from Gelfand's Trigonometry

Having a problem with an exercise from Gelfand and Saul's Trigonometry, in the section dealing with the half-angle formulae. The exercise (7.a. on p.151) asks the reader to show that: ...
2
votes
1answer
116 views

Is it true that $\forall b \forall c \forall x ((x^2 + bx + c \neq 0) \rightarrow b^2 - 4c < 0)$?

Well, I proved that $\forall b \forall c (b^2 - 4c \geq 0 \rightarrow \exists x(x^2 + bx + c = 0))$. This implies that $\forall b \forall c (\neg \exists x(x^2 + bx + c = 0) \rightarrow b^2 - 4c < ...
0
votes
1answer
104 views

Fitting a hyperboloid to 3 different radii

I would like to fit a hyperboloid to a set radii, but I must be making some mistake in solving for my derived constants. The question is technically only two-dimensional in nature, but I'm using a ...
6
votes
3answers
1k views

Find the equation which has 4 distinct roots

For which value of $k$ does the equation $x^4 - 4x^2 + x + k = 0 $ have four distinct real roots? I found this question on a standardized test, and the answer presumably relies on a graphing ...
1
vote
2answers
228 views

function defined by the average rate of change

Given a differentiable function $f(x)$, let $g(x)$ be defined by $g(x) = \begin{cases} (f(x)-f(a))/(x-a) &\mbox{if } x \neq a \\ f'(a) & \mbox{if } x = a. \end{cases}$ Suppose also f(x) is ...
0
votes
2answers
74 views

Is it possible to evaluate $\sum_{r=1}^{20}\frac{1}{r(r+1)}$ using $\sum_{r=1}^{n}\frac12n(n+1)$ [duplicate]

Evaluate $$\sum_{r=1}^{20}\frac{1}{r(r+1)}$$ It splits into $$\sum_{r=1}^{20}\frac{1}{r}-\sum_{r=1}^{20}\frac{1}{r+1}$$ I'm stuck on how to apply the standard result $\sum_{r=1}^{n}\frac12n(n+1)$ to ...
1
vote
3answers
82 views

Show that $\frac{1}{r!}-\frac{1}{(r+1)!}\equiv\frac{r}{(r+1)!}$.

Show that $\frac{1}{r!}-\frac{1}{(r+1)!}\equiv\frac{r}{(r+1)!}$. I get $$\frac{1}{r!}-\frac{1}{(r+1)!}=\frac{(r+1)-r!}{r!(r+1)!}$$ and in the numerator since $$(r+1)!-r!=r$$ so ...
1
vote
3answers
57 views

Use the identity $(r+1)^3-r^3\equiv3r^2+3r+1$ to find $\sum_{r=1}^nr(r+1)$

Use the identity $(r+1)^3-r^3\equiv3r^2+3r+1$ to find $$\sum_{r=1}^nr(r+1)$$ I can obtain $$\sum_{r=1}^n3r^2+3r+1=(n+1)^3-1$$ and I think the next step is ...
1
vote
2answers
77 views

Integer outputs of $y=x^2$ , do their last digits form an irrational?

Let the domain of $y=x^2$ be the positive integers. I input consecutive positive integers from $[1, \infty)$ their last digits are $a, b, c, ...$ respectively. If I then make the number $z=\frac ...
1
vote
1answer
57 views

Truck and packet problem

Ten trucks, numbered 1 to 10 , are carrying packets of sugar. Each packet weighs either 999 g or 1000 g and each truck carries only the packets of equal weights. The combined weight of 1 packet ...
1
vote
1answer
50 views

Sum of largest two angles

All the inner angles of a 7 sided polygon are obtuse, their sizes in degrees being distinct integers divisible by 9. What is the sum (in degree) of the largest two angles?
3
votes
1answer
115 views

Is there any kind of irrational number wich does not contain digit 9?

At first we must prove that there is or is`t irrational numbers which does not contain digit 9! if there are many kind of such numbers, then there is another question: how to write down algebraic ...
5
votes
3answers
616 views

property of real number system

"Between every two rational numbers there exist infinite irrational numbers and between every two irrational numbers there exist infinite rational numbers. Is this statement correct? If it is, then ...
1
vote
0answers
61 views

Roots of A Non-linear Equation

I have the following non-linear equation $$b_1\left(\frac{1}{f_1^2}-\frac{1}{(f_1-a_1)^2}\right)=b_2\left(\frac{1}{f_2^2}-\frac{1}{(f_2-a_2)^2}\right)$$ where $$f_1+f_2=A(\ \mbox{constant})$$ When I ...
0
votes
3answers
338 views

how to find out any digit of any irrational number?

We know that irrational number has not periodic digits of finite number as rational number. All this means that we can find out which digit exist in any position of rational number. But what about ...
4
votes
2answers
160 views

Solving an equation arising from method of image charges

I'm currently studying electrodynamics where the following equation arose: $0 = \frac{q}{\sqrt{R^2 + d^2 - 2Rd\cos\theta}} + \frac{q_p}{\sqrt{R^2 + d_p^2 - 2Rd_p\cos\theta}}$ where I need to solve ...
11
votes
3answers
1k views

How to solve : $\,8^x=6x$

I am stuck on the following problem which one of my friends gave me: Solve : $\,8^x=6x$. MY ATTEMPTS: We see that $$8^x=6x \implies 2^{3x}=6x.$$ Now I am not sure how to proceed further. ...
3
votes
3answers
187 views

Confusion about meaning of this question. High school Algebra level.

The product of two numbers is 10. One of them is $a$. Express their sum in terms of $a$. The factors of 10 are 1,2,5,10. Thus $a \in {1,2,5,10}$ Thus the sum can be either 7 or 11. Now how ...
1
vote
2answers
3k views

What is a standard precalculus syllabus?

I'm about to start teaching a calculus I class next week and I was wondering what I can expect from my students. I'm a Brit teaching in the US so I am unfamiliar with the system. I am hoping that ...
0
votes
2answers
96 views

Bringing numbers in a set closer together

I'm trying to express the relative scale of items compared to an "average" scale for all items in the set. This produces values like { 0.5, 1.8, 0.3, 1.2 ... } I then wish to use this number to ...
-1
votes
1answer
158 views

Paying less interest rate in credit card

I think I've found a way to pay off my credit cards faster, and therefore paying less interest rate. There's a Google Spreadsheet with my work at http://goo.gl/NqliZM My question What payment ...
-3
votes
2answers
1k views

Egg problem-Brain Teaser-Amazon Interview Question

A lady from the chicken farm gathers the eggs and brings it to sell it in the market. She sells the eggs but few eggs are left over. The 2nd day the left over eggs was doubled. Yet she sells the ...