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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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 $x$, ...
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
94 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 ...
49
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
157 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 ...
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! :)
3
votes
3answers
84 views

Why does $n \geq 2$ imply that $\frac n 2 < n$?

It has been a while since I did math proof in school, and I just can't figure out why $$n \geq 2 \text{ implies that } \frac n 2 < n$$ Anything would help! Thanks.
4
votes
6answers
119 views

Prove that $ 1+2q+3q^2+…+nq^{n-1} = \frac{1-(n+1)q^n+nq^{n+1}}{(1-q)^2} $

Prove: $$ 1+2q+3q^2+...+nq^{n-1} = \frac{1-(n+1)q^n+nq^{n+1}}{(1-q)^2} $$ Hypothesis: $$ F(x) = 1+2q+3q^2+...+xq^{x-1} = \frac{1-(x+1)q^x+xq^{x+1}}{(1-q)^2} $$ Proof: $$ P1 | F(x) = \frac{1-(...
0
votes
0answers
21 views

How would you solve an equation in two variables multiplied between each other?

In particular, how would you solve such an equation: $$ \alpha(1-\beta)x_2-\beta(1-\alpha)x_1+(\beta-\alpha)x_1x_2=0$$ with $0<\alpha<\beta<1$ and $x_1,x_2\in[0,2]$. Obviously one solution is ...
0
votes
1answer
20 views

Deal-Grove Model: How do we arrive at $F_{1}=F_{2}=F_{3}=F=\frac{C^{*}}{\frac{1}{k_{s}}+\frac{X_{ox}}{D}+\frac{1}{h}}$

Seems like a trivial question, but I'm confused. What is the step by step way to combine three equations in the Deal-Grove model: $$F_{1}= h(C^{*}-C_{o})$$ $$F_{2}= \frac{D(C_{o}-C_{i})}{X_{ox}}$$ $$...
0
votes
1answer
39 views

Parameterise linear combination of cosines

How do I parameterise the following implicit surface? $$ \cos x + \cos y + \cos z = 0 $$ Motivation for this problem comes from attempting to find stable motion for an object balanced on one point. ...
1
vote
2answers
50 views

Expanding a factorial

Can you explain me how we got this identity? $$\frac{1}{(3n)!}$$ the same as $$\frac{(3n)!}{(3n+3)!}$$ I have been trying to expand, but didn't get the same. Thanks.
0
votes
1answer
18 views

Determine all intervals of numbers $x$ satisfying the following inequalities.

i) $(x-5)^2 (x+10)\leq 0$ ii) $(x-5)^4 (x+10) \leq 0$ My answer : i) $(-10)\leq x \leq (5)$. ii) $(-10)\leq x \leq (5)$. Can you check my answer?
0
votes
0answers
18 views

Solving a cubic equation by factorization

I have the cubic equation $G^3 - (C - C_0)G - \frac{F}{\eta}$, with the constants $C, C_0, F, \eta \in \Re$. I want to find the solutions, so I tried factorising the equation, but I have after ...
0
votes
1answer
36 views

Volume of an igloo

AN igloo is a hemispherical structure built of blocks of ice by the Eskimos. THe interior diameter of the floor of an igloo is 6. Explain whether an Eskimo who is 1.7m tall could stand up straight ...
1
vote
4answers
59 views

If $9 ≥ 4x + 1$, which inequality represents the possible range of values of $12x + 3?$

If $9 ≥ 4x + 1$, which inequality represents the possible range of values of $12x + 3$? I've been trying to do SAT prep, and I came across this question. It allowed me to show an explanation and it ...
0
votes
0answers
28 views

How to return unique values by adding 5 variables

In the below image, I need to replace the Return values with the most appropriate sets of numbers so that when I add one Type, one Sub-Type, 0-2 User options, and the age together, I get the most ...
0
votes
1answer
27 views

Factorising an expression with two variables.

If the function $f(x) = 3x^2 + 5xy -2y^2 - 5x + 4y + p$ can be expressed as a product of two linear functions, find p. Hence express $f(x)$ as a product of two linear functions.
-1
votes
0answers
62 views

Solve $x*(x^x)=a$ for $x$

I tried with Lambert W-function but it seems it can not solve it. $x*x^x = a$ $a$ is constant find $x=f(a)$
2
votes
4answers
127 views

Common tangents to circle $x^2+y^2=\frac{1}{2}$ and parabola $y^2=4x$

I'm having trouble with this. What i do is say $\epsilon: y=mx+b$ is the tangent and it meets the circle at $M_1(x_1,y_1)$, i equate the $y$ of the tangent with the circle: $y=\pm \sqrt{1/2-x^2}$ and ...
0
votes
1answer
22 views

Exponential decay + a recurrence relation

I'm not sure if I get this right, some pointers could be helpful. Say you have to take 60m of some sort of medication at midnight. It has a blood half-life of 6 hours. Meaning that after 24 hours 3....
4
votes
2answers
88 views

How would you find the roots of $x^3-3x-1 = 0$

I'm not too sure how to tackle this problem. Supposedly, the roots of the equation are $2\cos\left(\frac {\pi}{9}\right),-2\cos\left(\frac {2\pi}{9}\right)$ and $-2\cos\left(\frac {4\pi}{9}\right)$ ...
0
votes
2answers
75 views

Solve for y in $2^{y + 3} = 5^{y}$

Solve for y in $2^{y + 3} = 5^{y}$. I know that \begin{align} \begin{split} 2^{y + 3} = 5^{y} \\ (y + 3) log 2 = y log 5 \\ \frac{y+3}{y} =\frac{log 5}{log2} \end{split} \end{align} and then I got ...