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.

learn more… | top users | synonyms (2)

2
votes
1answer
38 views

Prove that $f(x+z)$ has $4$ roots $\pm \alpha$ and $\pm \beta$

Let $a$ be a real parameter such that $$f_a(x)= x^4-6x^3+11ax^2-3(2a^2+3a-3)x+1$$ has has four distinct complex roots, that form a parallelogram when plotted on the Argand diagram. Prove That $...
4
votes
3answers
114 views

Why is finding the roots of a polynomial equation so important? What is to gain? [duplicate]

I have just started a pre calculus class, and our first lessons have been reviews on polynomial equation, quadratics and finding roots or solutions to equations. The topic is fairly simple but I just ...
0
votes
1answer
49 views

Exponential Quadratic Equation $-3\left(\frac{2}{3}\right)^x + 2 = x^2-2$ [closed]

How do you solve this equation? $-3\left(\frac{2}{3}\right)^x + 2 = x^2-2$. I have no clue how to do this and any help would be appreciated.
0
votes
1answer
24 views

Solve Graphically

Solve the given systems of equations by graphical method: $$x^2+y^2=5$$ and $$y=2x$$ My Attempt Let's have a look at the second equation ; $$y=2x$$ This is a linear equation in two variables ...
0
votes
0answers
55 views

How do I rearrange this equation without *any* form of $x$ on the other side?

It's easy to solve for $x$ in this equation: $$y=x-f(x)-1$$ where $f(x)$ is a different function than y, but I need to solve for $x$ without $f(x)$ on the other side. How would I accomplish this? ...
1
vote
1answer
67 views

If $\frac{a}{b}=\frac{b}{c}=\frac{c}{d}$, then prove that $(a^2+b^2+c^2)(d^2+b^2+c^2)=(ab+bc+cd)^2$

It is given that $\frac{a}{b}=\frac{b}{c}=\frac{c}{d}$. So how will I prove that $(a^2+b^2+c^2)(d^2+b^2+c^2)=(ab+bc+cd)^2$?
7
votes
2answers
93 views

How to arrive at Ramanujan's nested radicals?

Ramanujan found that $\sqrt[3]{\cos\left(\frac {2\pi}{7}\right)}+\sqrt[3]{\cos\left(\frac {4\pi}{7}\right)}+\sqrt[3]{\cos\left(\frac {8\pi}{7}\right)}=\sqrt[3]{\frac {1}{2}\left(5-3\sqrt[3]{7}\right)}$...
0
votes
1answer
6 views

Show that $x(1- (\frac{R_1-R_2}{x})^{2})^{0.5}$ = x – $\frac{(r1 – r2)^2}{2x}$

Show that $x(1- (\frac{R_1-R_2}{x})^{2})^{0.5}$ = x [1 – {0.5 $\frac{(r1 – r2)}{x}$ + … ] = x – $\frac{(r1 – r2)^2}{2x}$ by using binomial theorem it was mentioned that binomial theorem is $(x+y)^2$ ...
0
votes
1answer
20 views

Entropy, stirling's approximation

Given that the $\Omega$ is the total theoretical information encoded in a string of characrs let's say, we can express it in the bit-manner: $$\Omega=2^G$$ Therefore $G=\log_2 \Omega$ We know that $...
1
vote
3answers
47 views

Changing $y=mx+b$ equation into $ax+by=c$

I'm stuck on this question and I'm not totally sure how to transform an equation of the form $y=mx+b$ into an equation of the form $ax+by=c$. This is how far I have gotten $$y=-\frac{1}{3}x+\frac{29}...
1
vote
1answer
77 views

approximate irrational numbers by rational numbers

I want to prove this below: (1) For any irrational number $\alpha$, there exist infinitely many rational numbers $\frac{m}{n}$ such that $\left| {\alpha - \frac{m}{n}} \right| < \frac{1}{{{n^2}}}$...
0
votes
0answers
32 views

What is the meaning of the negative solution of the pythagorean theorem?

If $a^2=b^2+c^2$, then $a=\pm\sqrt{b^2+c^2}$ 1) What is the meaning of the negative solution for $a$? 2) How could I visualize that? 3) Similar question for the law of cosines. What is the meaning ...
0
votes
2answers
100 views

Proving that $\pi$ and $e$ are rational numbers [duplicate]

Maybe this question is too dumb to be asked, but it's really bugging me so I decide to ask it anyway. I hope you bear with me. Okay, it's known that both sides of the following series equal. $$\pi=...
0
votes
1answer
34 views

A bounded interval covered by finite open intervals

If a bounded closed interval $[a,b]$ is covered by finite open intervals $\bigcup\limits_{j = 1}^n {({c_j},{d_j})} $, I want to prove $b - a < \sum\limits_{j = 1}^n {({d_j} - {c_j})} $. It seems ...
4
votes
1answer
69 views

Ways to squeeze $e$ by hand

Let $a$ and $b$ be the lower and upper bound of $e$, respectively. Both $a$ and $b$ are rational numbers. Without using a calculator and without knowing the value of $e$, find $a$ and $b$ where $b-a&...
0
votes
1answer
37 views

Find value on scale of 0 to 1 including negative and positive numbers

I've been mulling this issue in my head for a few hours. I'm sure its quite simple, but I'm having trouble breaking exactly what I need to here. (Its been many years since my college days, and I don't ...
2
votes
1answer
76 views

How to solve this without rigorous calculation?

$${{(99!)^{100}\cdot 99-(98!)^{100}\cdot 98}\over {(97!)^{100}\cdot 97}}=?$$ This is the question. This is not MCQ , neither an answer is given . I have come to the following expression : $${{99^{...
8
votes
2answers
105 views

Does $\frac{x-2}{3x-6}$ really equal $\frac{1}{3}$?

In my maths lesson today we were simplifying fractions by factorising. One question was something like this: $\frac{x-2}{3x-6}$, which I simplified as $\frac{x-2}{3x-6}=\frac{x-2}{3(x-2)}=\frac{1}{3}$....
0
votes
3answers
53 views

How to get kids thinking about the relationship between basic math (algebra/precalculus) and English [closed]

I was wondering, how would you explain to young high school students or grade school kids the relationship between math and English? For me, both are systems that follow certain rules. Math is often ...
1
vote
2answers
74 views

Why is calculus normally taught after trigonometry (instead of more immediately after algebra)?

This question is a little meta. I hope I'm in the right place. In my experience the teaching of calculus is normally delayed until after learning basic trigonometry. Now that I've started learning ...
5
votes
0answers
107 views

Polynomials with degree $5$ solvable in elementary functions?

Quadratic, cubic and quartic polynomials are solvable in radicals, so there is no question here. What about the polynomials of degree $5$ (quintic)? Do we know all such polynomials (classes of ...
1
vote
0answers
13 views

Word for equivalence preserving transformations of equations

I am searching for a mathematical term describing an algebraic manipulation of an equation which preserves equivalence. So while adding $2$ to both sides of an equation results in an equivalent ...
12
votes
6answers
1k views

How do you simplify this square root of sum: $\sqrt{7+4\sqrt3}$?

I came around this expression when solving a problem. $$\sqrt{7+4\sqrt{3}}$$ WolframAlpha says it equals $2+\sqrt{3}$. We can confirm it like this $$\left(2+\sqrt{3}\right)^2 \;=\; 4+4\sqrt{3} + 3 ...
1
vote
2answers
51 views

Binomial Coefficient Identity Involving Summation

Prove that $$\sum_{j=0}^n (-1)^j \binom{n+j-1}{j}\binom{N+n}{n-j} = \binom{N}{n} $$ I tried to prove this via binomial expansions of $(1-x)^N (1+x)^{-m}$, and equating the coefficients of $x$, ...
-3
votes
0answers
29 views

Algebraic expresstions , word problem [closed]

A,B,C,D and E went to the restaurant for dinner. A paid half of the bill, B paid one-fifth of the bill, and rest of the bill paid by each?
4
votes
2answers
301 views

What is the value of $\frac{a^2}{b+c} + \frac{b^2}{a+c} + \frac{c^2}{a+b}$ if $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 1$?

If $$\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 1$$ then find the values of $$\frac{a^2}{b+c} + \frac{b^2}{a+c} + \frac{c^2}{a+b}.$$ How can I solve it? Please help me. Thank you in advance.
0
votes
3answers
61 views

How to calculate $(-\frac{1}{3} - 3)(-\frac{1}{3} + 5) - (-\frac{1}{3} + 4)(-\frac{1}{3} - 5)$

$(-\frac{1}{3} - 3)(-\frac{1}{3} + 5) - (-\frac{1}{3} + 4)(-\frac{1}{3} - 5)$ multiply and calculate left $(-\frac{1}{3} - 3)(-\frac{1}{3} + 5)$ = $-15\frac{5}{9}$ Same for right $(-\frac{1}{3} + 4)(...
6
votes
2answers
110 views

Show that $x^2 + y^2 + z^2 \ge 35$ if $x+3y+5z \ge 35.$

Show that $x^2 + y^2 + z^2 \ge 35$ if $x+3y+5z \ge 35.$ I have tried everything (proof by contradiction, etc.) but I can't seem to get it. The book didn't give any constraints whatsoever. Any hints ...
6
votes
1answer
88 views

Restricted equality involving prime numbers

Given three real numbers such that $a + b + c = 0$, it can be proved that \begin{align*} \frac{a^{5} + b^{5} + c^{5}}{5} & = \frac{a^{3} + b^{3} + c^{3}}{3}\cdot \frac{a^{2} + b^{2} + c^{2}}{2}\\ \...
1
vote
0answers
48 views

Trigonometric Roots of a Polynomial

After wondering on this question, I wondered how would you be able to find the roots of a polynomial, in the form $y=x^3+ax^2+bx+c$ if they are the sums of cosines? I'm wondering if it can, too, be ...
48
votes
9answers
6k views

Square root confusion: Why am I getting an answer if it doesn't work?

Alright, so I have $\sqrt{x-15} = 3-\sqrt{x}$. I first square both sides to get $x-15 = (3-\sqrt{x})(3-\sqrt{x})$ which simplifies to $x-15 = 9 -6\sqrt{x} + x$. I solved for $x$ and got $x = 16$, ...
0
votes
2answers
32 views

How to resolve this proportion/equivalence calculation? [simple one]

Let's suppose I have one cat and when buying food for him I have to take into account this: 1 cat eats 2kg of food each 20 days How can I get a formula to know how many days my food will last based ...
4
votes
2answers
156 views

How can I solve this hard system of equations?

Solve the system below \begin{align} &\sqrt {3x} \left( 1+\frac {1}{x+y} \right) =2\\ &\sqrt {7y} \left( 1-\frac{1}{x+y} \right) =4\sqrt{2} \end{align} Frankly I am disappointed, ...
9
votes
1answer
54 views

If $a$ and $b$ be the roots of the quadratic equation $x^2-6x+4=0$ then find the value of given expression.

Let $a$ and $b$ be the roots of the quadratic equation $x^2-6x+4=0$ and $P_n = a^n + b^n$ then the value of $$\frac{P_{50}(P_{48}+P_{49})-6P_{49}^2+4P_{48}^2}{P_{48}.P_{49}}$$ Options are $(A)$ $2$ ...
4
votes
2answers
127 views

Find $I$ in $\frac{\overline{SIX}}{\overline{NINE}}=\frac23$

In $\frac{\overline{SIX}}{\overline{NINE}}=\frac23$ every letter denotes a UNIQUE digit,find $I$. Expanding the fraction in base $10$ we have: $300S+30I+3X=2020N+200I+2E$ , but this doesn't ...
0
votes
4answers
55 views

When to simplify a quadratic equation?

I had the following quadratic equation: $$38x^2 - 140x - 250 = 0$$ And before starting to solve it, I simplified it by dividing all terms by $2$: $$19x^2 -70x - 125 = 0$$ But when I solved it I got: $...
0
votes
2answers
68 views

proving no real roots exist

Prove that $x^8-x^7+x^2-x+15$ has no real roots. I did it by first assuming it has real roots and then applying Descartes rule of signs. We find that if there are any real roots, they all must be ...
1
vote
0answers
50 views

What is meant by a function being linear in two variables?

I'm trying to understand the Mangasarian condition in the context of dynamic optimization (see here p 8.12) and am not sure what exactly is meant by a function $f(x,u)$ being linear in $x$ and $u$. If ...
2
votes
1answer
15 views

Proportion questioning

2 firms make the following charges for renting a car over the weekend . Firm A - Has a fixed charge of $320, and Charged 50 cents per km for every km over 300 Firm B - has a fixed charge of $60 and ...
1
vote
2answers
43 views

Proving $(w-1)^m$ is purely imaginary.

I'm having trouble trying to prove this: Let $ m\in \mathbb Z$, m even and $w\in\mathbb C$ a primitive $2m$-th root of unity. Prove that $(w-1)^m$ is purely imaginary. What I've tried to do so ...
10
votes
3answers
447 views

Why are there two versions of a polar equation for a circle from geometric form

In class today we learned that a rectangular/geometric equation for a circle such as $x^2+(y-5)^2 = 9$ can be converted into a polar equation by reducing it to the quadratic equation $r^2-10r\sin \...
2
votes
1answer
75 views

If $f(f(x)) = f(x^2)$, then must there be some constant $c$ such that $f(x)=c$ for all values of $x$ in the domain of $f$?

Here is a problem from Rusczyk-Crawford's Art of Problems Solving: Intermediate Algebra textbook (Chapter 2 Review, problem 2.30). If $f(f(x)) = f(x^2)$, then must there be some constant $c$ such ...
1
vote
1answer
19 views

Probability of Getting a Yahtzee of Fives Given Two Fives

(The following problem is from MAML, Meet 3, Round 1, December 2012, Problem 3.) In the game of Yahtzee one has a chance to get Yahtzee (5 of the same kind, such as 5 sixes) in the throw of 5 ...
4
votes
2answers
76 views

The sequence $(a_n)$ is given as $a_1=1, a_{2n} = a_n - 1, a_{2n+1} = a_n + 1$. $a_{2015}=$?

The sequence $(a_n)$ is given as $a_1=1, a_{2n} = a_n - 1, a_{2n+1} = a_n + 1$. What's the value of $a_{2015}$ Correct answer should be $a_{2015} = 9$. How? thing that came to mind was to see what $...
1
vote
5answers
62 views

Algebraic manipulation of a limit.

What are the algebraic manipulations and steps that makes the limit \begin{equation} \lim_{x\to2}\left(\frac{x^3-8}{x-2}\right) \end{equation} equal to \begin{equation} \lim_{x\to2}(x^2+2x+4) \end{...
1
vote
4answers
79 views

If $0 \le a \le 1$, then show that $xa + (1-a)y$ will always lie between $x$ and $y$.

If $0 \le a \le 1$, then show that $xa + (1-a)y$ will always lie between $x$ and $y$. I am sorry if this may seem like elementary question. I have tried many examples and they all seem to work. ...
2
votes
1answer
36 views

Plot of a function

What is the plot of: $$y=\frac{\beta(1-\alpha)x}{\alpha(1-\beta)+(\beta-\alpha)x}$$ with $0<\alpha<\beta<1$. How do I handle the parameters? How do I compute the derivatives to check for ...
3
votes
3answers
22 views

What is the logarithm of $(a-b)\delta_{ij}+b$?

Just now I came across the expression similar to: $x_{ij} = (a-b)\delta_{ij}+b$ The author then somehow converts this expression, into: $\ln x_{ij} = (\ln a-\ln b)\delta_{ij}+\ln b$ This comes ...
0
votes
2answers
42 views

Solution of irrational equations

I need some help solving these equations: $ \sqrt{2x+1} - \sqrt{x+8} > 3$ and $ \sqrt{3x^2 - 5a^2} = 2a - x$ Thank you in advance! :)