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
2k 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 ...
4
votes
3answers
281 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 ...
4
votes
5answers
4k 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 ...
0
votes
2answers
85 views

Proving the geometric sum formula by induction

$$\sum_{k=0}^nq^k = \frac{1-q^{n+1}}{1-q}$$ I want to prove this by induction. Here's what I have. $$\frac{1-q^{n+1}}{1-q} + q^{n+1} = \frac{1-q^{n+1}+q^{n+1}(1-q)}{1-q}$$ I wanted to factor a ...
48
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. $$\sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7\cdots}}}}}$$
27
votes
2answers
1k views

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

I'm currently studying exponential functions, however I've come across this before and it was just a law that I was told to accept without much proof or explanation, could someone explain the ...
61
votes
14answers
3k views

Why rationalize the denominator?

In grade school we learn to rationalize denominators of fractions when possible. We are taught that $\frac{\sqrt{2}}{2}$ is simpler than $\frac{1}{\sqrt{2}}$. An answer on this site says that "there ...
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 ...
11
votes
5answers
5k 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
401 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 ...
8
votes
6answers
622 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 ...
3
votes
5answers
2k views

Prove that $\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$

$$\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$$ I have attempted this question by expanding the left side using the cosine sum and difference formulas and then multiplying, and then ...
2
votes
4answers
2k views

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

How do I factor this polynomial: $2x^2-5xy-y^2$ ?
46
votes
10answers
2k 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 ...
21
votes
5answers
897 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? ...
7
votes
7answers
559 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.
5
votes
5answers
4k 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 ...
7
votes
12answers
13k views

Derivation of the formula for the vertex of a Parabola

I'm taking a course on Basic Conic Sections, and one of the ones we are discussing is of a parabola of the form $y = a x^2 + b x + c$ My teacher gave me the formula: $x = -\frac{b}{2a}$ as the $x$ ...
2
votes
1answer
155 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
296 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 ...
9
votes
5answers
694 views

Algorithms for “solving” $\sqrt{2}$

The very first words out of my mouth need to be this... "Solving" is the wrong term since I am speaking about irrational numbers. I just don't know which word is the correct word... So that can be ...
5
votes
4answers
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 ...
3
votes
4answers
2k views

Find an equation of the plane

I have come across a difficult problem in my textbook. The problem ask: Find an equation of the plane that passes through the line of intersection of the planes $x-z=1$ and $y+2z=3$ and is ...
244
votes
33answers
32k 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} ...
29
votes
11answers
4k views

How do you define functions for non-mathematicians?

I'm teaching a College Algebra class in the upcoming semester, and only a small portion of the students will be moving on to further mathematics. The class is built around functions, so I need to ...
40
votes
6answers
2k views

Is $\sqrt[3]{p+q\sqrt{3}}+\sqrt[3]{p-q\sqrt{3}}=n$, $(p,q,n)\in\mathbb{N} ^3$ solvable?

In this recent answer to this question by Eesu, Vladimir Reshetnikov proved that $$ \begin{equation} \left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1} \end{equation} $$ ...
66
votes
15answers
4k views

math fallacy problem: $-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1$?

I know there is something wrong with this but I don't know where. It's some kind of a math fallacy and it is driving me crazy. Here it is: $$-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1?$$
16
votes
2answers
6k 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 ...
8
votes
4answers
759 views

How does partial fraction decomposition avoid division by zero?

This may be an incredibly stupid question, but why does partial fraction decomposition avoid division by zero? Let me give an example: $$\frac{3x+2}{x(x+1)}=\frac{A}{x}+\frac{B}{x+1}$$ Multiplying ...
5
votes
2answers
411 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 ...
8
votes
4answers
691 views

How to detect when continued fractions period terminates

I'm doing continued fractions arithmetic. Is there a method to detect when a continued fractions period terminates? Let me give you an example: $\sqrt{2} = [1; \overline{2}]$, $\sqrt{7} = [2; ...
2
votes
2answers
1k 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 ...
3
votes
2answers
228 views

simplify $(a_1 + a_2 +a_3+… +a_n)^m$

How to simplify this best $(a_1 + a_2 +a_3+... +a_n)^m$ for $m=n, m<n, m>n$ I could only get $\sum_{i=0}^{m}\binom{m}{i}a_i^i\sum_{j=0}^{m-i}\binom{m-i}{j}a_j ... $
8
votes
2answers
968 views

Finding roots of polynomials with rational coefficients

I'm looking for a general approach (or approaches) for finding the roots of polynomials with rational coefficients of higher degrees than $4$. The problem is that I need to find the exact roots and ...
3
votes
2answers
131 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 ...
3
votes
4answers
769 views

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ show that $x=-c/b$ when $a=0$

OK, this one has me stumped. Given that the solution for $ax^2+bx+c =0$ $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\qquad(*)$$ How would you show using $(*)$ that $x=-c/b$ when $a=0$ (Please dont use $a=0$ ...
0
votes
3answers
154 views

Finding the range of $f(x) = 1/((x-1)(x-2))$

I want to find the range of the following function $$f(x) = \frac{1}{(x-1)(x-2)} $$ Is there any way to find the range of the above function? I have found one idea. But that is too critical. Please ...
4
votes
4answers
119 views

Finding range of $\frac{x}{(1-x)^2}$

I'm working though some exercises. One of them is asking to find the range of $f(x) = \frac{x}{(1-x)^2}$. The chapter this exercise belongs to is after the one where the differentiation power rule is ...
4
votes
2answers
636 views

How can I compute $\sum\limits_{k = 1}^n \frac{1} {k + 1}\binom{n}{k} $?

This sum is difficult. How can I compute it, without using calculus? $$\sum_{k = 1}^n \frac1{k + 1}\binom{n}{k}$$ If someone can explain some technique to do it, I'd appreciate it. Or advice using ...
1
vote
6answers
477 views

How could we solve $x$, in $|x+1|-|1-x|=2$?

How could we solve $x$, in $|x+1|-|1-x|=2$? Please suggest a analytical way that I could use in other problems too like this $ |x+1|+|1-x|=2$ and of this genre. Thank you,
22
votes
3answers
618 views

Why everytime the final number comes the same?

I have come across an interesting puzzle. Write $20$ numbers. Erase any two number say $x$ and $y$ and and replace with $\text{Number}_{new} = xy/(x + y)$ OR $\text{Number}_{new}= x + y + xy$ ...
6
votes
3answers
959 views

Challenging inequality: $abcde=1$, show that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}+\frac{33}{2(a+b+c+d+e)}\ge{\frac{{83}}{10}}$

Let $a,b,c,d,e$ be positive real numbers which satisfy $abcde=1$. How can one prove that: $$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} +\frac{1}{e}+ \frac{33}{2(a + b + c + d+e)} ...
14
votes
4answers
567 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 ...
14
votes
1answer
390 views

Why does Group Theory not come in here?

Here is a list of questions that I find quite similar, for the one and only reason that they all show much "symmetry". Which is at the same time my problem, because I don't have a very precise notion ...
14
votes
2answers
2k views

What is the best way to solve an equation involving multiple absolute values?

An absolute value expression such as $|ax-b|$ can be rewritten in two cases as $|ax-b|=\begin{cases} ax-b & \text{ if } x\ge \frac{b}{a} \\ b-ax & \text{ if } x< \frac{b}{a} \end{cases}$, ...
8
votes
5answers
536 views

Proof that the equation $x^2 - 3y^2 = 1$ has infinite solutions for $x$ and $y$ being integers

I have seen the Pell's equation wiki page but I need to prove this from scratch without mentioning any formula. I have also seen multiple answers on this site but the answers tend to skip over and ...
11
votes
5answers
5k views

How can I write an equation that matches any sequence?

One thing I have been wondering about lately is how to write an equation that describes a pattern of numbers. What I mean is: x 0 1 2 y 1 5 9 If ...
17
votes
4answers
2k 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.
14
votes
1answer
3k views

Using Vieta's theorem for cubic equations to derive the cubic discriminant

Background: Vieta's Theorem for cubic equations says that if a cubic equation $x^3 + px^2 + qx + r = 0$ has three different roots $x_1, x_2, x_3$, then $$\begin{eqnarray*} -p &=& x_1 + x_2 ...