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)

1
vote
1answer
117 views

Solving $4t+{2\over t-3}+{2\over3}=0$ [duplicate]

Possible Duplicate: How would I rearrange this equation to make the subject $t$? The scenario is as follows: A sub-atomic particle is travelling in a straight line through a tubular ...
2
votes
5answers
447 views

What is the least value of the function?

What is the least value of function: $$y= (x-2) (x-4)^2 (x-6) + 6$$ For real values of $x$ ? Does $\frac{dy}{dx} = 0$, give the value of $x$ which will give least value of $y$? Thanks in ...
3
votes
3answers
246 views

Find the minimum value of $3x + 4y$

The minimum value of $3x + 4y$ subject to the condition $$x^2 y^3 = 6$$ and $x$ and $y$ are positive .
3
votes
2answers
3k views

Quartic Equation Solution and Conditions for real roots?

Q1. How to solve a Quartic Equation. There is an online calculator available (and many more similar) that gives the precise answers and also defines the method. Does anyone know what the source of ...
1
vote
1answer
177 views

Finding the minimum value of $\left(\frac{a + 1}{a}\right)^2 + \left(\frac{b + 1}{b}\right)^2 + \left(\frac{c + 1}{c}\right)^2 $

Find the minimum value of $$\left(\frac{a + 1}{a}\right)^2 + \left(\frac{b + 1}{b}\right)^2 + \left(\frac{c + 1}{c}\right)^2 $$ I tried to expand it and break it into individual terms and use ...
2
votes
2answers
151 views

Find the minimum value.

Find the minimum value of $4^x + 4^{1-x}$ , $x\in\mathbb{R}$. In this I used the property that $a + \frac{1}{a}\geq 2$. So I begin with $$ 4^x + \left(\frac{1}{4}\right)^x + ...
5
votes
4answers
447 views

Simplifying an expression

The following expression is given: $$\frac{x^7+y^7+z^7}{xyz(x^4+y^4+z^4)}$$ Simplify it, knowing that $x+y+z=0$.
0
votes
2answers
203 views

Simplifying exponents, multiplication, and addition

How can you get $10^{n+1}$ from $9\cdot 10^n+10^n$? This is part of a proof I am working on.
0
votes
1answer
110 views

Where can you perform variable substitution in an equation?

I've ended up in a debate on a proof for .999˜ being equal to 1 (in the real number system.) I think the arguments have boiled down to whether or not we can preform variable substitution on one side ...
1
vote
1answer
406 views

Mixture problem? Repeatedly extract/replace X units of fluid (with fluid #2) to attain X% mixture ratio?

Let's say I have a 100 liter container of water. I extract 10 liters of water, and replace it with 10 liters of juice. Now, the container is now 10% juice / 90% water. I can keep repeating this to ...
1
vote
1answer
66 views

Does $n^2+(n+k)^2+(n+2k)^2+ \ldots +(n+mk)^2$ have a general equation?

Does $n^2+(n+k)^2+(n+2k)^2+\ldots+(n+mk)^2$ have a general formula? e.g. $$1^2+2^2+3^2+\ldots+n^2=n(n+1)(2n+1)/6$$
3
votes
4answers
180 views

How to prove that $\frac{\cos\alpha}{\cos\beta}=a/b$

If $\alpha \not= \beta$, and $$ a\tan \alpha+b\tan\beta=(a+b)\tan\frac{\alpha+\beta}{2}$$ then can we prove that $\frac{\cos\alpha}{\cos\beta}=\frac{a}{b}$? Seems like I am stuck on this one.
0
votes
1answer
118 views

Prove if a polar function involves only the rational numbers and sin, cos, tan functions, it can be written in rectangular form.

Prove if a function only have including the rational numbers and sin, cos, tan function and $r$, you always could write it in rectangular form. Ex. For $r=2/(2+2\cos\theta)$ it could be represent in ...
0
votes
2answers
266 views

How to make this polynomial have equal roots?

I have the equation- $$(-3p)^2 + 4(4p+1)$$ how do I make it have equal roots? cause I don't think it's possible but must be, can someone please help
1
vote
1answer
137 views

I need a formula to estimate how fast a vehicle travels

I am looking for a formula to estimate how fast a vehicle travels down a dirt track with the following data: Jeep #$1$: $1000$ hp, weighs $1700$ pounds and travels $300$ ft. in $3.6$ seconds Jeep ...
8
votes
3answers
427 views

Solve $x^y \, = \, y^x$ [duplicate]

Possible Duplicate: $x^y = y^x$ for integers $x$ and $y$ I obtained a question asking for how to solve $\large x^y = y^x$. The given restraints was that $x$ and $y$ were both positive ...
1
vote
2answers
77 views

Is it possible to check if transformed function is correct using test points?

Most of the (basic) math I've been doing in school has been easily checkable - it is immediately obvious if one had the right answer by just plugging it back in. Recently, doing function ...
-1
votes
1answer
137 views

How to solve this non-linear system? Get the roots directly.

How to solve this non-linear system? Get the roots directly. $ax^3+bx^2+cx+d$ $\frac{b}{a}=-(\alpha+\beta+\gamma)$ $\frac{c}{a}=\alpha\beta+\beta\gamma+\gamma\alpha)$ ...
1
vote
1answer
38 views

Transforming points from function

Currently, I am doing transformations on functions, with things like $y=f(x+2)$ representing a transformation of the original function 2 to the left, for example. However, I am stuck on transforming ...
0
votes
1answer
156 views

Question about the diagonal of a square

The question states "Find the side of a square whose diagonal is 5 feet longer that its side". It seems easy but I'm not sure about my answer. Since I know that a square has equal sides, I assign $x$ ...
1
vote
1answer
64 views

Find the value of $A/B$?

$$A = x^4 ( 1 + y^4 ) + y^4 ( 1 + z^4 ) + z^4 ( 1 + x^4 )$$ $$B = x^2 y^2 z^2$$ If $x$, $y$ and $z$ are real, which of the following can be the value of $A/B$? Options i) 0 ii) 2 iii) 4 iv) 8
2
votes
1answer
135 views

What is a short way to deal with this cubic polynomial problem?

i tried to denote the roots to be $a,(a+1),b$ in this problem and i set up a bunch of equations but they are too complicated to solve. What is a way to solve this problem and check it within 3-4 ...
9
votes
2answers
682 views

Puzzle: The number of quadratic equations which are unchanged by squaring their roots is

My friend asked me this puzzle: The number of quadratic equations which are unchanged by squaring their roots is My answer is: 3 $x^2-(\alpha+\beta)x +\alpha\beta = 0$ where $\alpha$ and ...
2
votes
0answers
124 views

Euler Four-Square Identity variants?

Is it well-known that there are an infinite number of Euler-type 4-square identities? Proof: $\begin{align} &{(x_1^2+ x_2^2+ x_3^2+ x_4^2) (y_1^2+ y_2^2+ y_3^2+ y_4^2)\,=\,z_1^2+ z_2^2+ z_3^2+ ...
0
votes
1answer
302 views

Inverse of the function $f(x) = e^x + \arctan(x) = y$

Given the function $$ f(x) = e^x + \arctan(x) = y\;, $$ what is the inverse $f^{-1}(y)=\dots\;$, and how can I find it? I’m looking for solutions including all steps and possible explanations ...
2
votes
1answer
228 views

Number of integer solutions of $\frac{1}{x} + \frac{1}{y} = \frac{1}{1000}$

What is the number of integer solutions of: $$\frac{1}{x} + \frac{1}{y} = \frac{1}{1000}$$ How to solve these type of problems if am comfortable of solving $x+y=z$. But how to do if multiplicative ...
1
vote
1answer
1k views

Round Dimension to Rectangular Duct Dimension Formula

When calculating the diameter equivalent ($\text{de}$) of a round duct in a HVAC system, and we know the rectangular duct size, we can use the following formula: $$\text{de} = 1.30 \times \frac{(a ...
2
votes
3answers
1k views

The square root of a variable is negative?

If the square root of a variable is negative, as shown below: $$\sqrt x = -1$$ Then what is $x$ equal to? The closest answer I can think of is $i^4$. $$\sqrt{i^4}=i^{\frac42}=i^2=-1$$ But if $i^4$ ...
1
vote
2answers
5k views

How can I find the roots of a quadratic function?

Bascially we are trying to find the roots of a quadratic equation, and 'apparently' there is a theorem for this, but every one that I have found so far mentions that the degree of the polynomial is ...
6
votes
2answers
97 views

A question about divisibility.

What I've observed: Pick any $3$ random positive integers, say $a$, $b$, $c$ which are not of the form $0\pmod{3}$ then one and only one of $a+b$, $b+c$, $c+a$, $a+b+c$ is always a multiple of $3$. ...
1
vote
2answers
94 views

Find the value of a?

Let $a$ and $b$ be two positive numbers such that $a\gt b$. Let $G$ be the geometric mean of $a$ and $b$ (that is, $G=\sqrt{ab}$), and $H$ be the Harmonic mean of $a$ and $b$, that is, $$H = ...
0
votes
1answer
791 views

Find the common ratio of geometric progression

If $p,q$, and $r$ are terms of an arithmetic progression are also in a geometric progression, then find the common ratio of the geometric progression in terms of $p,q$, and $r$.
0
votes
1answer
167 views

Simplifying this equation (euler's formula?)

I was reading some stuff earlier today, and I wasn't sure how they changed the exponentials to trigs in this expression: ...
1
vote
4answers
70 views

Help with unknown expansion

I'm reading through a number theory text and the following equivalence is used in a proof. It looks kind of like a binomial expansion, but not. I don't understand why this is true. $a^n - 1= ...
1
vote
4answers
271 views

Solving the inequality $\frac{x}{\sqrt{x+12}} - \frac{x-2}{\sqrt{x}} > 0$

I'm having troubles to solve the following inequality.. $$\frac{x}{\sqrt{x+12}} - \frac{x-2}{\sqrt{x}} > 0$$ I know that the result is $x>0$ and $x<4$ but I cannot find a way to the ...
0
votes
1answer
310 views

What is the sum of the squares of roots of the equation?

What is the sum of the squares of roots of the equation below? $$m^{1/3} + ( 2m-3)^{1/3} = \big(12 ( m – 1)\big)^{1/3}$$
2
votes
3answers
207 views

Why does $ \sum_{i=1}^{n}\frac{1}{n} = 1$?

I have come across the following in my Computational Finance slides but I'm unsure on the final equality. $$ \bar{r}_P = \sum_{i=1}^{n}w_i\bar{r_i} = \sum_{i=1}^{n}\frac{1}{n}\bar{r} = \bar{r}$$
1
vote
1answer
72 views

Why are we adding instead of subtracting?

The problem statement: Distance between two stations A and B is 230 km. Two motorcyclist starts simultaneously from A and B in opposite directions and the distance between them after 4 hours ...
0
votes
1answer
85 views

How to solve for x

$W(4d^2 (1-x^2)^2) = abc^3x \sqrt{(\pi^2 (i-x^2)^2 + 16 x^2) }$ I have to find x ,i have the values of all other constants , I tried to separate it using partial fraction but I am stuck. a=3 b=4 c=7 ...
4
votes
2answers
115 views

Completing the square for $2x - x^2$

How do you complete the square for $2x-x^2$?
1
vote
3answers
172 views

What makes $0!$ equal to 1? [duplicate]

Possible Duplicate: Prove $0! = 1$ from first principles I don't understand how it's equal to 1. Also, I found that $(-x)!$ is equal to complex $\infty$. How is this so?
2
votes
2answers
120 views

What is this formula?

There is some formula that I can't precisely remember for polynomials, which goes something like $x^n-1 = (x-1)(\text{a lot of stuff})$. It could be more general, like $x^n - k$, or maybe it is just ...
5
votes
2answers
2k views

Solving the equation $- y^2 - x^2 - xy = 0$

Ok, this is really easy and it's getting me crazy because I forgot nearly everything I knew about maths! I've been trying to solve this equation and I can't seem to find a way out. I need to find out ...
3
votes
1answer
70 views

Prove or name this identity: the number of factors of $p$ in $\binom{n}{k}$ is $(s_p(k)+s_p(n-k)-s_p(n))/(p-1)$

you can count the number of factors of $p$ that are in $\binom{n}{k}$ for prime $p$. Let $s_p(n)$ be the sum of the digits of $n$ in base $p$. Then, the number of factors of $p$ in $\binom{n}{k}$ is ...
0
votes
1answer
73 views

The graph of $-5y=10$

Let's say I have this equation: $-5y=10$. How would I graph that on a piece of graphing paper? Thanks
0
votes
1answer
372 views

The diophantine equation $a+b+c+d+e = abcde$

Now you may think that I am annoying, but if I am not asking this question, then it seems not so complete and I can't grasp the whole idea... refer to this question: Positive rationals satisfying ...
1
vote
2answers
126 views

Simple question about powers and radicals

I had a question in my calculus book that was written a little strangely and I tried to rewrite it to be more simple but it did not work out. It made me realize something about powers and radicals ...
1
vote
1answer
49 views

How to show that $GCD(f,f')=(x-a_1)^{r_1-1}\cdots(x-a_l)^{r_l-1}$

Given any polynomial $f\in \mathbb C[x]$ of degree $n>0$, $f$ can be written in the form $f=c(x-a_1)^{r_1}\cdots(x-a_l)^{r_l},$ where $a_1,...,a_l$ are distinct. Also, $f'$ is the product ...
4
votes
1answer
386 views

How to find all rational numbers satisfy this equation?

Find all rational number $a,b,c$ satisfy: $$a+b+c=abc$$ I try to change this in different forms like $(ab-1)c = a+b$, $(ac-1)b = a+c$, $(cb-1)a = b+c$ etc but it won't help...
2
votes
1answer
83 views

confused about order of operations

Ok, I'm confused on which is the next step in solving this equation: $$0 = (6.0\text{ kg}) (-3.0\frac{\text{m}}{\text{sec}}) + (78\text{ kg}) v$$ I'm supposed to solve for $v$, however, I'm not ...