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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19
votes
6answers
3k views

Show that the $\max{ \{ x,y \} }= \frac{x+y+|x-y|}{2}$.

Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$. I do not understand how to go about completing this problem or even where to start.
12
votes
7answers
2k views

Derivation of the general forms of partial fractions

I'm learning about partial fractions, and I've been told of 3 types or "forms" that they can take (1) If the denominator of the fraction has linear factors: $${5 \over {(x - 2)(x + 3)}} \equiv {A ...
5
votes
5answers
3k views

Steps to solve this system of equations: $\sqrt{x}+y=7$, $\sqrt{y}+x=11$

I want to solve this system of equations, I have been out of Maths for a long!! $$\sqrt{x}+y=7$$ $$\sqrt{y}+x=11$$ Just wondering easiest step to find values for $x$ and $y$ from the above ...
3
votes
1answer
247 views

Equation with high exponents

I would appreciate any help with this problem: $ x^8+2x^7+2x^6+5x^5+3x^4+5x^3+2x^2+2x^1+1x^0=0 $ I know that when $x$ isn't zero $x^0=1$ so the equation could be re-written as $ ...
29
votes
10answers
4k views

Direct Proof that $1 + 3 + 5 + \cdots+ (2n - 1) = n\cdot n$

Prove that for all integers $n$, $n \geq 1$, $$1 + 3 + 5 + \cdots + (2n - 1) = n\cdot n$$ How would I go about proving this?
28
votes
2answers
2k views

Why is ${x^{\frac{1}{2}}}$ the same as $\sqrt x $?

Why is ${x^{\frac{1}{2}}}$ the same as $\sqrt x $? I'm currently studying indices/exponents, and this is a law that I was told to accept without much proof or explanation, could someone explain ...
23
votes
5answers
1k views

How does one actually show from associativity that one can drop parentheses?

I've always heard this reasoning, and it makes obvious sense, but how do you actually show it for some arbitrary product? Would it be something like this? ...
22
votes
13answers
7k views

How to solve $4\sin \theta +3\cos \theta = 5$

Another problem that I already wasted hours on. Given $$4\sinθ +3\cosθ = 5$$ Find $$4\cosθ -3\sinθ$$ Help me guys (PS:I'm not that good in maths)
3
votes
2answers
13k views

Range scaling problem

I have a few ranges which I want to scale but I'm missing the formula (and common sense). For example I have a scale range from 40 to 100, but I want my data to range from 0 - 100. What formula do I ...
11
votes
6answers
7k views

Factorize the polynomial $x^3+y^3+z^3-3xyz$

I want to factorize the polynomial $x^3+y^3+z^3-3xyz$. Using Mathematica I find that it equals $(x+y+z)(x^2+y^2+z^2-xy-yz-zx)$. But how can I factorize it by hand?
9
votes
1answer
322 views

Find the sum : $\frac{1}{\cos0^\circ\cos1^\circ}+\frac{1}{\cos1^\circ \cos2^\circ} +\frac{1}{\cos2^\circ \cos3^\circ}+…+$

Find the sum of the following : (i) $$\frac{1}{\cos0^\circ \cos1^\circ}+\frac{1}{\cos1^\circ\cos2^\circ} +\frac{1}{\cos2^\circ \cos3^\circ}+......+\frac{1}{\cos88^\circ \cos89^\circ}$$ I tried : ...
14
votes
3answers
608 views

Is there any “superlogarithm” or something to solve $x^x$? [duplicate]

Is there any "superlogarithm" or something to solve an equation like this: $$x^x = 10?$$
6
votes
5answers
294 views

Given that $x^y=y^x$, what could $x$ and $y$ be?

It's not too difficult to figure out that $x$ and $y$ can both be 1, and also $x$ can be 2 and $y$ can be 4 (and vice versa). But I can't rule out if there are other solutions. Does it have anything ...
5
votes
5answers
5k views

Simple Proof by induction: “9 divides $n^3 + (n+1)^3 + (n+2)^3$”

I'm trying to prove using induction that 9 divides $n^3 + (n+1)^3 + (n+2)^3$ whenever $n$ is a non-negative integer. So far, I have: Base case: P(1) = (1) + (8) + (27) = 36, 36 can be divided by 9 ...
271
votes
33answers
34k views

Pedagogy: How to cure students of the “law of universal linearity”?

One of the commonest mistakes made by students, appearing at every level of maths education up to about early undergraduate, is the so-called “Law of Universal Linearity”: $$ \frac{1}{a+b} ...
22
votes
3answers
9k views

Strategies to denest nested radicals.

I have recently read some passage about nested radicals, I'm deeply impressed by them. Simple nested radicals $\sqrt{2+\sqrt{2}}$,$\sqrt{3-2\sqrt{2}}$ which the later can be denested into ...
36
votes
7answers
2k views

Is there any geometric way to characterize $e$?

Let me explain it better: after this question, I've been looking for a way to put famous constants in the real line in a geometrical way -- just for fun. Putting $\sqrt2$ is really easy: constructing ...
25
votes
8answers
2k views

Is there a name for this strange solution to a quadratic equation involving a square root?

Here's an elementary question on solving the following quadratic equation (well, it's not a quadratic until the square root is eliminated): $$\sqrt{x+5} + 1 = x$$ Upon solving the above equation ...
17
votes
4answers
720 views

How to prove $\sum\limits_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$?

Other than the general inductive method,how could we show that $$\sum_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$$ Apart from induction,I tried with Wolfram Alpha to check the validity,but ...
12
votes
5answers
6k views

Is there any formula for the series $1 + \frac12 + \frac13 + \cdots + \frac 1 n = ?$

Is there any formula for this series? $$1 + \frac12 + \frac13 + \cdots + \frac 1 n .$$
8
votes
3answers
600 views

Simplifying $\sqrt[4]{161-72 \sqrt{5}}$

$$\sqrt[4]{161-72 \sqrt{5}}$$ I tried to solve this as follows: the resultant will be in the form of $a+b\sqrt{5}$ since 5 is a prime and has no other factors other than 1 and itself. Taking this ...
6
votes
5answers
354 views

Proving the AM-GM inequality for 2 numbers $\sqrt{xy}\le\frac{x+y}2$

I am having trouble with this problem from my latest homework. Prove the arithmetic-geometric mean inequality. That is, for two positive real numbers $x,y$, we have $$ \sqrt{xy}≤ \frac{x+y}{2} .$$ ...
4
votes
3answers
294 views

I need some advice for $ \; “ \; 3xy+y-6x-2=0 \;”\, $

PS: I edited the title again and I think, it's better now... :) Actually, almost the whole solving way is wrong but, I understood why... :D :) You can check that or add some more useful links about ...
52
votes
10answers
3k views

What Is Exponentiation?

Is there an intuitive definition of exponentiation? In elementary school, we learned that $$ a^b = a \cdot a \cdot a \cdot a \cdots (b\ \textrm{ times}) $$ where $b$ is an integer. Then later on ...
9
votes
4answers
84k views

How to figure out the log of a number without a calculator?

I have seen people look at log (several digit number) and rattle off the first couple of digits. I can get the value for small values (aka the popular or easy to know roots), but is there a formula. ...
13
votes
2answers
463 views

Question Regarding Cardano's Formula

In Cardano's derivation of a root of the cubic polynomial $f(X)=X^3+bX+c$ he splits the variable $X$ into two variables $u$ and $v$ together with the relationship that $u+v=X$. From this he finds that ...
8
votes
6answers
682 views

Is it wrong to say $ \sqrt{x} \times \sqrt{x} =\pm x,\forall x \in \mathbb{R}$?

Is it wrong to say $$ \sqrt{x} \times \sqrt{x} =\sqrt{x^2}= \pm x$$ I am quite sure that $\sqrt{(x)^2} = \pm(x)$ But, does $\sqrt{x } \times \sqrt{x} =- (x)$ doesn't holds in $\mathbb{R}$ but if we ...
12
votes
7answers
1k views

Solve trigonometric equation: $1 = m \; \text{cos}(\alpha) + \text{sin}(\alpha)$

Dealing with a physics Problem I get the following equation to solve for $\alpha$ $1 = m \; \text{cos}(\alpha) + \text{sin}(\alpha)$ Putting this in Mathematica gives the result: $a==2 ...
6
votes
5answers
422 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( ...
4
votes
2answers
786 views

Polynomial $p(a) = 1$, why does it have at most 2 integer roots?

The question that I am trying to answer is : Suppose is $p(x)$ is a polynomial with integer coefficients. Show that if $p(a) = 1$ for some integer a then $p(x)$ has at most two integer roots. I have ...
3
votes
4answers
1k views

Formula for calculating $\sum_{n=0}^{m}nr^n$

I want to know the general formula for $\sum_{n=0}^{m}nr^n$ for some constant r and how it is derived. For example, when r = 2, the formula is given by: $\sum_{n=0}^{m}n2^n = 2(m2^m - 2^m +1)$ ...
3
votes
7answers
613 views

Proof of the formula $1+x+x^2+x^3+ \cdots +x^n =\frac{x^{n+1}-1}{x-1}$ [duplicate]

Possible Duplicate: Value of $\sum x^n$ Proof to the formula $$1+x+x^2+x^3+\cdots+x^n = \frac{x^{n+1}-1}{x-1}.$$
2
votes
4answers
2k views

How to factor the quadratic polynomial $2x^2-5xy-y^2$?

How do I factor this polynomial: $2x^2-5xy-y^2$ ?
8
votes
5answers
2k views

Geometric mean never exceeds arithmetic mean

This was a mathematical induction question proposed in a textbook, and I've exhausted multiple approaches (proving RHS - LHS > 0, splitting the fraction, fractional exponents, etc.) The geometric ...
5
votes
8answers
3k views

How to solve $\sin x +\cos x = 1$?

No matter how I do it, I always end up with $x = 0, 90, 270$ and $360$. All of those except $270$ is right, but I can't quite figure out how to get the $270$ degrees out of the answer. I've tried ...
2
votes
6answers
213 views

Prove the inequality $|xy|\leq\frac{1}{2}(x^2+y^2)$

How can I prove the inequality $|xy|\leq\frac{1}{2}(x^2+y^2)$ I have tried substitute $x,y$ for numbers, which turns out right, but I don't know how to reason here. Thanks in advance!
22
votes
7answers
711 views

Product of cosines: $ \prod_{r=1}^{7} \cos \left(\frac{r\pi}{15}\right) $

Evaluate $$ \prod_{r=1}^{7} \cos \left({\dfrac{r\pi}{15}}\right) $$ I tried trigonometric identities of product of cosines, i.e, $$\cos\text{A}\cdot\cos\text{B} = \dfrac{1}{2}[ ...
29
votes
9answers
6k views

How are first digits of $\pi$ found?

Since Pi or $\pi$ is an irrational number, its digits do not repeat. And there is no way to actually find out the digits of $\pi$ ($\frac{22}{7}$ is just a rough estimate but it's not accurate). I am ...
16
votes
5answers
885 views

Fibonacci number identity.

How do I see that $f_{n+1}f_{n-1} = f_n^2 + (-1)^n$, $n \ge 2$, where $f_1 = 1$, $f_2 = 1$, and $f_{n+2} = f_{n+1} + f_n$ for $n \in \mathbb{N}$?
20
votes
4answers
3k views

Can a finite sum of square roots be an integer?

Can a sum of a finite number of square roots of integers be an integer? If yes can a sum of two square roots of integers be an integer? The square roots need to be irrational.
4
votes
2answers
2k views

Create polynomial coefficients from its roots

Given some roots : $r_1,r_2,\ldots,r_n$, how can we reconstruct polynomial coefficients? I know the Horner scheme and that we can just go backwards receiving those coefficients. But I'm curious if ...
12
votes
6answers
22k views

Can the square root of a real number be negative?

Can the square root of a real number be negative? Dealing with the questions of functions in eleventh class my maths teacher says that square root of a real number is always positive. How is it ...
9
votes
8answers
624 views

What is the result of $\lim\limits_{x \to 0}(1/x - 1/\sin x)$?

Find the limit: $$\lim_{x \rightarrow 0}\left(\frac1x - \frac1{\sin x}\right)$$ I am not able to find it because I don't know how to prove or disprove $0$ is the answer.
8
votes
3answers
1k views

Evaluation of $\int\frac{\sqrt{\cos 2x}}{\sin x}\,dx$

Compute the indefinite integral $$ \int\frac{\sqrt{\cos 2x}}{\sin x}\,dx $$ My Attempt: $$ \begin{align} \int\frac{\sqrt{\cos 2x}}{\sin x}\,dx &= \int\frac{\cos 2x}{\sin^2 x\sqrt{\cos ...
8
votes
6answers
7k views

How to prove $(f \circ\ g) ^{-1} = g^{-1} \circ\ f^{-1}$?

I'm doing exercise on discrete mathematics and I'm stuck with question: If $f:Y\to Z$ is an invertible function, and $g:X\to Y$ is an invertible function, then the inverse of the composition $(f ...
9
votes
3answers
3k views

Direct formula for area of a triangle formed by three lines, given their equations in the cartesian plane.

I read this formula in some book but it didn't provide a proof so I thought someone on this website could figure it out. What it says is: If we consider 3 non-concurrent, non parallel lines ...
5
votes
4answers
7k views

Number of surjective functions from A to B

Am I on the right track? I am not sure about my reasoning... Number of surjective functions from $A$ to $B$ $$A = \{1,2,3,4\} ; B = \{a,b,c\}$$ We must count the surjective functions, meaning the ...
3
votes
2answers
145 views

find extreme values of $\frac{2x}{x²+4}$

I am doing my homework like a good little boy. I know that when I want to find the extreme values of a function I have to put the derivative equal to zero so I can find the x values. I've done it ...
2
votes
1answer
173 views

How to simplify $\sum_{i=1}^{k}\binom{n + i - 1}{i}$? [duplicate]

How to simplify $\sum_{i=1}^{k}\binom{n + i - 1}{i}$? I tried reducing the sum to $\binom{n}{1}, \binom{n}{2}, \binom{n}{3}$ and so on but couldn't get a pattern.
1
vote
1answer
352 views

Establishing formula from recurrence

Can anyone tell me how do we establish a formula from a given recurrence relation? Take the example of $f(n) = 2f(n-1) + 1$, $n \in \mathbb{Z^+}$, $f(1) = 1$ When I write out the first few values, it ...